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PROFESSOR: All right, I guess
we can get started here.

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So welcome.

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00:00:24,670 --> 00:00:28,290
Today we're going to do
Bayesian updating, which

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is to say we're going to
be using Bayes' Theorem

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00:00:30,490 --> 00:00:31,560
to learn from data.

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00:00:31,560 --> 00:00:33,200
We're going to
update our beliefs

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about various hypotheses
based on the data.

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For this week, we're going
to have known priors.

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00:00:40,190 --> 00:00:42,570
We're going to know our
belief about the data ahead

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00:00:42,570 --> 00:00:43,960
of time for certain.

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00:00:43,960 --> 00:00:50,480
We'll draw coins out of
drawers or pull things out

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00:00:50,480 --> 00:00:52,020
of hats, things like that.

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00:00:52,020 --> 00:00:54,120
Starting next week,
and we discussed

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this a little last
time, we'll have

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00:00:56,340 --> 00:00:58,050
unknown priors,
which is you'll have

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00:00:58,050 --> 00:00:59,380
to be making up those priors.

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00:00:59,380 --> 00:01:03,050
And we'll teach you
techniques for making it up.

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00:01:03,050 --> 00:01:10,360
So the first slide is just
XKCD's view of Bayes' Theorem.

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00:01:10,360 --> 00:01:10,860
Let's go on.

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00:01:10,860 --> 00:01:14,645
We want to start with a
clicker question here.

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00:01:17,920 --> 00:01:21,027
So the question is: which
treatment would you choose

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00:01:21,027 --> 00:01:22,110
if you needed a treatment?

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00:01:22,110 --> 00:01:25,870
Treatment 1 cured
100% of the patients.

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00:01:25,870 --> 00:01:29,060
Treatment 2 cured
95% in a trial.

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00:01:29,060 --> 00:01:31,930
And treatment 3 cured 90%.

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00:01:31,930 --> 00:01:34,835
So with this information,
which would you choose?

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00:01:39,229 --> 00:01:40,770
AUDIENCE: With only
this information?

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00:01:40,770 --> 00:01:41,853
PROFESSOR: With only this.

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00:01:41,853 --> 00:01:44,610
If this is all I told you.

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00:01:44,610 --> 00:01:46,020
Which would you choose?

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00:01:46,020 --> 00:01:48,785
I recognizing that you might
not be too happy with just

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00:01:48,785 --> 00:01:52,430
this information, but if this
is their best information, what

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00:01:52,430 --> 00:01:53,590
would you choose?

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00:01:53,590 --> 00:01:57,110
All right, there's some
holdouts for the 95% cure.

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00:01:57,110 --> 00:02:00,470
I'm not sure why.

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00:02:00,470 --> 00:02:02,740
Really, that 95%
was just, you know,

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00:02:02,740 --> 00:02:08,139
they've cured one
person 19 out of 20--

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00:02:08,139 --> 00:02:11,640
they got them 19 out of 20
well, or however you say that.

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00:02:11,640 --> 00:02:12,390
Yeah.

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00:02:12,390 --> 00:02:16,880
All right, so there's
even a few more holdouts.

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00:02:16,880 --> 00:02:20,790
All right, what if I gave
you this information?

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00:02:20,790 --> 00:02:24,590
Treatment 1 cured 3
out of 3 patients.

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00:02:24,590 --> 00:02:26,430
That's 100%.

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00:02:26,430 --> 00:02:28,990
Treatment 2 cured
19 of 20 patients.

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00:02:28,990 --> 00:02:32,226
That's 95%.

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00:02:32,226 --> 00:02:33,600
Or, you have a
standard treatment

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which has cured 90,000 out
of 100,000 patients, 90%,

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00:02:38,920 --> 00:02:41,120
in clinical practice.

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00:02:41,120 --> 00:02:42,490
Now which one would you choose?

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00:02:49,832 --> 00:02:51,290
PROFESSOR 2: It's
very interesting.

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00:02:53,706 --> 00:02:54,747
Yeah, it totally changed.

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00:02:54,747 --> 00:02:55,670
PROFESSOR: What's that?

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00:02:55,670 --> 00:02:56,580
PROFESSOR 2: It totally shifted.

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00:02:56,580 --> 00:02:58,360
PROFESSOR: It totally
shifted, really?

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00:02:58,360 --> 00:03:00,734
PROFESSOR 2: We have a bunch
of very conservative people.

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00:03:00,734 --> 00:03:02,400
PROFESSOR: Right.

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So the majority would
choose the third choice,

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00:03:06,800 --> 00:03:09,720
90 out of 100,000.

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00:03:09,720 --> 00:03:13,554
I think-- well, someone tell
me, what's the intuition there?

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AUDIENCE: Because
it's been more tested.

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00:03:15,330 --> 00:03:16,705
PROFESSOR: It's
been more tested.

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3 out of 3 is 100%, but do
you really believe 100%?

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Not yet.

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00:03:21,300 --> 00:03:23,590
Maybe it'll prove out in time.

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19 out of 20 is a little
more, but for most people,

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for 79% of you,
that's not enough.

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00:03:31,620 --> 00:03:36,110
The 90 out of 100,000
seems like good odds,

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00:03:36,110 --> 00:03:37,900
and it's well tested.

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19 out of 20, if the next person
tested doesn't get better,

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00:03:44,130 --> 00:03:46,570
now you back to 90%.

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So that doesn't seem
like quite enough.

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00:03:48,650 --> 00:03:53,372
What if were 95 out
of 100 patients.

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00:03:53,372 --> 00:03:54,580
We'll let you click in there.

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00:03:54,580 --> 00:03:57,020
Change number 2 to 95
out of 100 and let's

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00:03:57,020 --> 00:04:00,370
see what people would do.

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00:04:00,370 --> 00:04:02,780
Oh, you're very
obliging, 95 out of 100--

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00:04:02,780 --> 00:04:04,230
PROFESSOR 2: Now
it's about 50-50.

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00:04:04,230 --> 00:04:05,527
PROFESSOR: About 50-50.

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00:04:05,527 --> 00:04:06,735
PROFESSOR 2: Between 2 and 3.

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00:04:09,250 --> 00:04:10,750
PROFESSOR:
Realistically, of course,

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00:04:10,750 --> 00:04:11,833
what would you want to do?

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00:04:11,833 --> 00:04:14,240
You'd want to do a
little more research

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00:04:14,240 --> 00:04:17,730
and find out what these trials
were, how good experiments you

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00:04:17,730 --> 00:04:20,820
thought they were, what
people are thinking about it.

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00:04:20,820 --> 00:04:23,290
But with just this data, that's
what people would choose.

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00:04:23,290 --> 00:04:25,040
And would people would
choose differently.

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00:04:25,040 --> 00:04:27,360
OK.

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00:04:27,360 --> 00:04:29,810
PROFESSOR 2: Let's see.

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00:04:29,810 --> 00:04:33,390
So now I'm going to give
you a toy problem that

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00:04:33,390 --> 00:04:35,670
actually mirrors the
same kind of effect we

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00:04:35,670 --> 00:04:37,470
saw on the last slide.

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00:04:37,470 --> 00:04:44,100
So, suppose in this MIT mug,
I have dice of two types,

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00:04:44,100 --> 00:04:47,710
4-sided and 20-sided.

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00:04:47,710 --> 00:04:48,210
All right?

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00:04:48,210 --> 00:04:50,760
So I'm going to
reach into the mug.

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00:04:50,760 --> 00:04:53,380
I'm not going to look.

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00:04:53,380 --> 00:04:55,880
Much.

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00:04:55,880 --> 00:04:58,935
I'm going to randomly
pull out a die.

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00:04:58,935 --> 00:04:59,810
I'm going to roll it.

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00:05:02,750 --> 00:05:05,200
All right, I got a 1.

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00:05:05,200 --> 00:05:06,450
OK.

109
00:05:06,450 --> 00:05:07,270
I got a 1.

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00:05:07,270 --> 00:05:13,190
Now, just based on this
information, which type of die

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00:05:13,190 --> 00:05:14,480
do you think I randomly chose?

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00:05:17,340 --> 00:05:17,962
The 4-sided.

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00:05:17,962 --> 00:05:20,170
Someone want to give an
explanation of why they think

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00:05:20,170 --> 00:05:21,728
the 4-sided is more likely?

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00:05:21,728 --> 00:05:23,307
AUDIENCE: It's
greater probability.

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00:05:23,307 --> 00:05:24,890
PROFESSOR 2: It's
greater probability.

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00:05:24,890 --> 00:05:28,180
What's the probability of
getting a 1 on the 4-sided die?

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00:05:28,180 --> 00:05:30,230
And what about a 20-sided sided?

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00:05:30,230 --> 00:05:30,960
OK.

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00:05:30,960 --> 00:05:35,300
So we see that it would
be more likely to roll a 1

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00:05:35,300 --> 00:05:38,180
with a 4-sided die than
with a 20-sided die.

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00:05:38,180 --> 00:05:43,570
So, suppose I tell you that
I really in this cup only

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00:05:43,570 --> 00:05:46,820
have one die of each type.

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00:05:46,820 --> 00:05:49,380
So does that change
your analysis

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00:05:49,380 --> 00:05:50,820
of which die is most likely?

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00:05:50,820 --> 00:05:52,680
Don't hit forward yet.

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00:05:52,680 --> 00:05:54,120
PROFESSOR: I would think of it.

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00:05:54,120 --> 00:05:56,450
My hands are staying.

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00:05:56,450 --> 00:06:00,130
PROFESSOR 2: Or do you want to
stick with the same reasoning?

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00:06:00,130 --> 00:06:00,850
All right.

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00:06:00,850 --> 00:06:02,683
Let me show you what's
actually in this cup.

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00:06:09,002 --> 00:06:09,960
PROFESSOR: Ouch.

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00:06:09,960 --> 00:06:12,630
PROFESSOR 2: Oh, you
can't see it yet.

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00:06:12,630 --> 00:06:15,720
Let's go with the
document camera.

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00:06:18,940 --> 00:06:22,610
OK, that was actually
what I had in this cup.

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00:06:22,610 --> 00:06:24,560
Now what?

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00:06:24,560 --> 00:06:26,510
So does this change at
all of your analysis

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00:06:26,510 --> 00:06:29,250
of which die you think I
rolled and got a 1 with?

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00:06:29,250 --> 00:06:31,082
What type of die, rather?

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00:06:31,082 --> 00:06:31,665
AUDIENCE: Yes.

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00:06:31,665 --> 00:06:32,100
PROFESSOR 2: Yeah?

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00:06:32,100 --> 00:06:33,810
Now what do you
think is more likely?

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00:06:33,810 --> 00:06:34,817
4-sided or 20-sided?

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00:06:34,817 --> 00:06:36,400
AUDIENCE: How many
20 sided are there?

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00:06:36,400 --> 00:06:38,441
PROFESSOR 2: I think I
got 20 20-sided die there.

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00:06:45,110 --> 00:06:49,080
All right, so raise your hand
if you like a 4-sided die.

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00:06:49,080 --> 00:06:51,850
Raise your hand if you
like the 20-sided die.

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00:06:51,850 --> 00:06:52,350
Great.

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00:06:52,350 --> 00:06:55,230
So that reasoning that
you're doing intuitively

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00:06:55,230 --> 00:06:57,560
is something that we want
to formalize this week

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00:06:57,560 --> 00:06:59,930
and next week using
Bayes' Theorem

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00:06:59,930 --> 00:07:02,460
to do these calculations
in a systematic way,

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00:07:02,460 --> 00:07:06,380
to evaluate hypotheses
based on data.

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00:07:06,380 --> 00:07:10,230
And these numbers,
for example, how many

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00:07:10,230 --> 00:07:12,370
die I had of one
type versus another,

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00:07:12,370 --> 00:07:14,980
we can encode that information
in what's called a prior,

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00:07:14,980 --> 00:07:16,285
and we'll come to that today.

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00:07:16,285 --> 00:07:18,410
And you should've seen, in
your reading, this idea.

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00:07:22,612 --> 00:07:24,820
PROFESSOR: OK so I'm going
to work through an example

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00:07:24,820 --> 00:07:25,420
to start with.

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00:07:25,420 --> 00:07:31,090
We have 3 types of coins:
type A, B, and C. However

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00:07:31,090 --> 00:07:34,210
it happens, the probability
of getting heads with type A

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00:07:34,210 --> 00:07:35,040
is 0.5.

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00:07:35,040 --> 00:07:36,130
That's a fair coin.

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00:07:36,130 --> 00:07:40,820
The probability of getting
heads with type B is 0.6,

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00:07:40,820 --> 00:07:44,000
and of type C is 0.9.

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00:07:44,000 --> 00:07:48,210
And here's a box that
contains two of the fair coins

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00:07:48,210 --> 00:07:49,460
and one of each of the other.

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00:07:49,460 --> 00:07:54,070
So two of type A, one of
type B, and one of type C.

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00:07:54,070 --> 00:07:56,490
Suppose I pick one at
random and I get heads.

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00:07:56,490 --> 00:08:00,160
So this is like John picking a
die at random, and getting a 1.

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00:08:00,160 --> 00:08:02,900
I get heads.

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00:08:02,900 --> 00:08:04,900
Here's the question:
before I flipped it,

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00:08:04,900 --> 00:08:08,930
I have a prior belief
in the probability

175
00:08:08,930 --> 00:08:11,290
that the coin was
of each type based

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00:08:11,290 --> 00:08:13,610
on the number in my drawer.

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00:08:13,610 --> 00:08:17,390
After flipping it,
my belief changes.

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00:08:17,390 --> 00:08:22,180
The probability will change
based on the data I get.

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00:08:22,180 --> 00:08:24,160
And so a question you
could ask: what was

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00:08:24,160 --> 00:08:27,590
learned by flipping the coin?

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00:08:27,590 --> 00:08:31,150
What we want to teach you to
do to answer these questions,

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00:08:31,150 --> 00:08:33,270
we have a nice
way of summarizing

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00:08:33,270 --> 00:08:36,280
all of that once in a table.

184
00:08:36,280 --> 00:08:44,090
So here the notation, we
have this nice scripty H--

185
00:08:44,090 --> 00:08:44,775
that's no good.

186
00:08:48,110 --> 00:08:49,496
H is our hypothesis.

187
00:08:52,820 --> 00:08:58,740
That is, you're of
type A or B or C.

188
00:08:58,740 --> 00:09:06,690
We're going to make a conjecture
on what we think the coin was.

189
00:09:06,690 --> 00:09:11,330
Before we do anything,
we have a prior.

190
00:09:11,330 --> 00:09:13,910
So what's the
probability of-- how

191
00:09:13,910 --> 00:09:17,240
did I label this-- that
the coin was of type A?

192
00:09:21,240 --> 00:09:24,045
Where all you know is I
pulled it out of the drawer.

193
00:09:24,045 --> 00:09:26,385
So what's the probability
it's of type A.

194
00:09:26,385 --> 00:09:28,135
PROFESSOR 2: Let me
go back and remind you

195
00:09:28,135 --> 00:09:31,570
what the drawer looks like.

196
00:09:31,570 --> 00:09:33,850
What is it?

197
00:09:33,850 --> 00:09:35,040
PROFESSOR: 0.5.

198
00:09:35,040 --> 00:09:36,590
0.5, or I'll write it like this.

199
00:09:36,590 --> 00:09:39,390
2 out of 4, which is 0.5.

200
00:09:39,390 --> 00:09:41,240
And what's the
probability-- I'll

201
00:09:41,240 --> 00:09:52,180
shorten this-- for just
type B. Yeah, 1/4 or 0.25.

202
00:09:52,180 --> 00:09:58,940
The probability that we're
of type C is 1/4 or 0.25.

203
00:09:58,940 --> 00:10:01,565
This was one of our
simpler counting problems.

204
00:10:04,130 --> 00:10:06,380
In the table, you can
see that's encoded

205
00:10:06,380 --> 00:10:09,860
in the line in the
column marked prior.

206
00:10:09,860 --> 00:10:13,150
Before we take any data,
this is what we'd say.

207
00:10:13,150 --> 00:10:16,221
This is the sort of
odds we'd give if we

208
00:10:16,221 --> 00:10:18,440
were going to bet on this.

209
00:10:18,440 --> 00:10:19,770
But now we have some data.

210
00:10:22,790 --> 00:10:28,000
So once we have data we
can get the likelihood.

211
00:10:32,470 --> 00:10:34,540
The likelihood is
the probability

212
00:10:34,540 --> 00:10:37,556
of seeing the data
given a hypothesis.

213
00:10:40,200 --> 00:10:43,590
I won't write this out, I'll
just point at the slide.

214
00:10:43,590 --> 00:10:46,749
John has a-- you were going
to make your mouse bigger.

215
00:10:46,749 --> 00:10:47,540
PROFESSOR 2: I was.

216
00:10:47,540 --> 00:10:48,123
PROFESSOR: OK.

217
00:10:50,790 --> 00:10:53,180
So the likelihood
column-- we use

218
00:10:53,180 --> 00:10:56,210
this term when we are talking
about maximum likelihood--

219
00:10:56,210 --> 00:11:00,240
is the probability of the
data given the hypothesis.

220
00:11:00,240 --> 00:11:06,010
And this is typically easy,
or often easy, to figure out.

221
00:11:06,010 --> 00:11:07,800
What's the probability
of seeing heads--

222
00:11:07,800 --> 00:11:12,850
our data was I flipped a heads--
if my coin is a 0.5 probability

223
00:11:12,850 --> 00:11:13,350
coin.

224
00:11:17,560 --> 00:11:25,550
So the probability of seeing
heads given a type A coin

225
00:11:25,550 --> 00:11:26,170
is what?

226
00:11:26,170 --> 00:11:28,660
Somebody.

227
00:11:28,660 --> 00:11:30,644
It's 0.5.

228
00:11:30,644 --> 00:11:32,310
This is particularly
easy because that's

229
00:11:32,310 --> 00:11:33,018
what we told you.

230
00:11:33,018 --> 00:11:38,480
The probability of heads
with a type A coin is 0.5.

231
00:11:38,480 --> 00:11:47,160
What about for B
and C. 0.6 and 0.9.

232
00:11:47,160 --> 00:11:52,650
So there right here is the
likelihood column in the table.

233
00:11:52,650 --> 00:11:56,350
Now, how do we
get the posterior?

234
00:11:56,350 --> 00:12:02,390
Bayes' Theorem says
the probability of H

235
00:12:02,390 --> 00:12:04,760
given D, which is what I want.

236
00:12:04,760 --> 00:12:06,630
I want to know,
given this data, what

237
00:12:06,630 --> 00:12:08,670
do I believe the coin to be?

238
00:12:08,670 --> 00:12:11,060
What's its probability?

239
00:12:11,060 --> 00:12:17,610
Is the probability of D given
H times the probability of H

240
00:12:17,610 --> 00:12:20,840
all over the probability of D.

241
00:12:20,840 --> 00:12:24,230
Well, notice this
numerator right here.

242
00:12:24,230 --> 00:12:28,430
That's just our likelihood
times our hypothesis,

243
00:12:28,430 --> 00:12:32,280
our prior belief
in our hypothesis.

244
00:12:32,280 --> 00:12:36,390
And so what do we get when
I multiply the prior column

245
00:12:36,390 --> 00:12:37,860
by the likelihood column?

246
00:12:37,860 --> 00:12:39,720
I get this column in red.

247
00:12:39,720 --> 00:12:41,910
And I'll call that the
unnormalized posterior.

248
00:12:41,910 --> 00:12:42,880
Why unnormalized?

249
00:12:42,880 --> 00:12:46,320
Because it doesn't sum to 1.

250
00:12:46,320 --> 00:12:48,160
How do I get it to sum to 1?

251
00:12:48,160 --> 00:12:52,070
There I have to divide by
the probability of the data.

252
00:12:52,070 --> 00:12:54,050
And how do I get the
probability of the data?

253
00:12:58,190 --> 00:13:03,942
That's called the-- I
see someone mouthing it.

254
00:13:03,942 --> 00:13:05,150
The Law of Total Probability.

255
00:13:07,920 --> 00:13:10,540
And the Law of Total
Probability says

256
00:13:10,540 --> 00:13:14,330
you multiply, for
each hypothesis,

257
00:13:14,330 --> 00:13:19,340
this value times this
value and add them up.

258
00:13:19,340 --> 00:13:25,820
Which is in fact just the
sum of the red column.

259
00:13:25,820 --> 00:13:28,340
So there I've written it: 0.625.

260
00:13:28,340 --> 00:13:29,750
And now, how do I normalize?

261
00:13:29,750 --> 00:13:34,160
I just divide all my values
in my unnormalized column

262
00:13:34,160 --> 00:13:37,430
by that normalizing
factor, 0.625,

263
00:13:37,430 --> 00:13:40,640
and I get these, what
I call the posterior.

264
00:13:44,690 --> 00:13:47,117
Let's say-- well, now
let's analyze this.

265
00:13:47,117 --> 00:13:48,200
So now what's your belief?

266
00:13:48,200 --> 00:13:52,130
Which coin is the
most likely coin?

267
00:13:52,130 --> 00:13:53,400
After seeing this data.

268
00:13:56,760 --> 00:13:59,870
Still A. But what's
happened to the probability

269
00:13:59,870 --> 00:14:02,770
from our prior probability?

270
00:14:02,770 --> 00:14:03,370
It's dropped.

271
00:14:03,370 --> 00:14:06,041
Which one has
gained probability.

272
00:14:06,041 --> 00:14:06,540
AUDIENCE: C.

273
00:14:06,540 --> 00:14:07,998
PROFESSOR: C. Does
that make sense?

274
00:14:07,998 --> 00:14:14,440
If I flip a heads, which coin is
the most likely to flip heads?

275
00:14:14,440 --> 00:14:15,930
C. So I've shifted it.

276
00:14:15,930 --> 00:14:19,120
Suppose I flipped
heads three more times.

277
00:14:19,120 --> 00:14:21,955
What would you see in the
probability as I kept updating?

278
00:14:29,763 --> 00:14:30,846
PROFESSOR 2: Say it again.

279
00:14:30,846 --> 00:14:32,290
AUDIENCE: The probability
of C gets bigger.

280
00:14:32,290 --> 00:14:33,664
PROFESSOR 2: The
probability of C

281
00:14:33,664 --> 00:14:36,800
would get bigger at the
expense of the probabilities

282
00:14:36,800 --> 00:14:37,830
of the others.

283
00:14:37,830 --> 00:14:39,740
If I flipped heads
9 times in a row,

284
00:14:39,740 --> 00:14:46,120
I'd be very convinced
that I had a type C coin.

285
00:14:46,120 --> 00:14:49,240
One last thing to
note about this table.

286
00:14:49,240 --> 00:14:50,920
I put totals down here.

287
00:14:50,920 --> 00:14:53,540
What does this total of 1
represent in the first column?

288
00:14:56,190 --> 00:14:56,690
Emily.

289
00:14:56,690 --> 00:14:59,445
AUDIENCE: The total
probabilities of anything, so

290
00:14:59,445 --> 00:15:01,120
the total space.

291
00:15:01,120 --> 00:15:02,800
PROFESSOR: Yeah.

292
00:15:02,800 --> 00:15:04,760
When I sum up the
total probabilities,

293
00:15:04,760 --> 00:15:07,200
they sum up to 1.

294
00:15:07,200 --> 00:15:08,440
Same with this last.

295
00:15:08,440 --> 00:15:10,030
That's a probability.

296
00:15:10,030 --> 00:15:12,390
This I could sum, because
they were probabilities.

297
00:15:12,390 --> 00:15:14,550
Why didn't I sum this column?

298
00:15:20,980 --> 00:15:22,670
Those are the
likelihoods, and what's

299
00:15:22,670 --> 00:15:26,010
changing as I move
down this column?

300
00:15:26,010 --> 00:15:30,570
What's changing is H. That's
not a probability function.

301
00:15:30,570 --> 00:15:32,970
It doesn't sum to 1.

302
00:15:32,970 --> 00:15:37,540
There's no reason that the
likelihoods have to sum to 1,

303
00:15:37,540 --> 00:15:42,200
and here you can see they
clearly don't sum to 1.

304
00:15:42,200 --> 00:15:43,330
So I don't sum that.

305
00:15:43,330 --> 00:15:48,390
That's an important, maybe
slightly subtle point,

306
00:15:48,390 --> 00:15:54,110
that the likelihood function
is not a probability function.

307
00:15:54,110 --> 00:15:55,410
OK, any questions here?

308
00:15:58,644 --> 00:16:00,500
PROFESSOR 2: All right.

309
00:16:00,500 --> 00:16:06,760
So, let's go back to
this document camera.

310
00:16:06,760 --> 00:16:08,540
I got my trusty platonic solids.

311
00:16:14,680 --> 00:16:18,810
So, we're going to go through a
series of board questions now,

312
00:16:18,810 --> 00:16:22,640
three board questions which are
going to allow you to practice

313
00:16:22,640 --> 00:16:25,240
making these tables and
getting a sense for how

314
00:16:25,240 --> 00:16:29,470
to do Bayesian Updating, how to
iterate it, just how it works.

315
00:16:29,470 --> 00:16:33,780
And our toy problem is going
to be involving these 5

316
00:16:33,780 --> 00:16:34,800
platonic dice.

317
00:16:34,800 --> 00:16:37,567
You all have them, but let's
not break them out today,

318
00:16:37,567 --> 00:16:39,900
because it would be very noisy
with the cameras I think.

319
00:16:39,900 --> 00:16:46,440
We'll use our 4-sided--
here, I got big ones.

320
00:16:46,440 --> 00:16:49,690
We've got our 4-sided,
8-sided, 6-sided.

321
00:16:49,690 --> 00:16:50,880
No 6-sided.

322
00:16:50,880 --> 00:16:53,546
No, I don't have a 6-sided.

323
00:16:53,546 --> 00:16:54,910
I have the wrong dice.

324
00:16:54,910 --> 00:16:55,800
Hold on.

325
00:16:55,800 --> 00:17:02,290
We got 20, we got 4, we
got 12, we got 8, ah-- 10,

326
00:17:02,290 --> 00:17:03,816
what are you doing there?

327
00:17:03,816 --> 00:17:04,649
That's not platonic.

328
00:17:09,800 --> 00:17:12,829
OK, and 6.

329
00:17:15,670 --> 00:17:17,760
So now I'm actually
going to take these.

330
00:17:17,760 --> 00:17:18,609
You can watch this.

331
00:17:18,609 --> 00:17:20,380
All right, this is empty cup.

332
00:17:24,079 --> 00:17:25,680
OK, they're in there right?

333
00:17:25,680 --> 00:17:26,180
OK.

334
00:17:28,950 --> 00:17:30,710
How do we do this?

335
00:17:30,710 --> 00:17:31,780
How do we convince them?

336
00:17:31,780 --> 00:17:32,810
PROFESSOR: Should we
choose one at random?

337
00:17:32,810 --> 00:17:33,560
PROFESSOR 2: Yeah.

338
00:17:36,985 --> 00:17:37,860
All right, choose it.

339
00:17:37,860 --> 00:17:40,580
Don't let anyone see.

340
00:17:40,580 --> 00:17:41,810
Hand it me.

341
00:17:41,810 --> 00:17:43,270
OK.

342
00:17:43,270 --> 00:17:45,100
So I have this die.

343
00:17:45,100 --> 00:17:47,190
Don't look at the others.

344
00:17:47,190 --> 00:17:50,300
I'm going to start
rolling it, all right?

345
00:17:50,300 --> 00:17:51,450
In this secret bin.

346
00:17:51,450 --> 00:17:53,700
Jerry, you can ver-- we can
have a student come verify

347
00:17:53,700 --> 00:17:56,200
that I'm not making this up.

348
00:17:56,200 --> 00:17:57,640
You ready?

349
00:17:57,640 --> 00:17:59,596
OK, here we go.

350
00:17:59,596 --> 00:18:01,360
OK, what did I get?

351
00:18:01,360 --> 00:18:02,220
8.

352
00:18:02,220 --> 00:18:03,200
All right.

353
00:18:03,200 --> 00:18:05,430
So, which die is it so far?

354
00:18:05,430 --> 00:18:06,340
What do you think?

355
00:18:06,340 --> 00:18:08,340
Could it be 4, the 4-sided die?

356
00:18:08,340 --> 00:18:09,580
6-sided?

357
00:18:09,580 --> 00:18:10,921
8?

358
00:18:10,921 --> 00:18:12,262
12?

359
00:18:12,262 --> 00:18:13,480
20?

360
00:18:13,480 --> 00:18:15,460
What's most likely?

361
00:18:15,460 --> 00:18:17,940
All right.

362
00:18:17,940 --> 00:18:20,522
Come take a look at another one.

363
00:18:20,522 --> 00:18:21,022
Ready?

364
00:18:25,802 --> 00:18:27,264
Ah, got a 1.

365
00:18:27,264 --> 00:18:28,430
AUDIENCE: It's the same die?

366
00:18:28,430 --> 00:18:30,910
PROFESSOR 2: It's the same
die, it's the same die.

367
00:18:30,910 --> 00:18:33,061
All right, so now what?

368
00:18:33,061 --> 00:18:35,470
How does this change,
anyone want to tell me

369
00:18:35,470 --> 00:18:40,450
how would this, now, change from
before what you believe to be?

370
00:18:40,450 --> 00:18:43,060
Does it change what you
believe at all, in any way?

371
00:18:43,060 --> 00:18:44,930
You still think
it's the 8-sided.

372
00:18:44,930 --> 00:18:48,350
Does it make the 8-sided less
likely, more likely relative

373
00:18:48,350 --> 00:18:49,880
to the others?

374
00:18:49,880 --> 00:18:50,690
More likely.

375
00:18:50,690 --> 00:18:51,190
Wonderful.

376
00:18:51,190 --> 00:18:53,090
See you guys all sense
this in your bones.

377
00:18:53,090 --> 00:18:53,690
It's great.

378
00:18:53,690 --> 00:18:54,340
This is going to be so easy.

379
00:18:54,340 --> 00:18:55,340
For all of you.

380
00:18:55,340 --> 00:18:56,870
All right.

381
00:18:56,870 --> 00:19:00,310
We've got to keep going until
we figure out with this die.

382
00:19:00,310 --> 00:19:01,630
You, you love to participate.

383
00:19:01,630 --> 00:19:02,180
Come on.

384
00:19:05,376 --> 00:19:07,740
Ready?

385
00:19:07,740 --> 00:19:08,240
AUDIENCE: 3.

386
00:19:08,240 --> 00:19:09,621
PROFESSOR 2: 3, oh man.

387
00:19:09,621 --> 00:19:10,120
All right.

388
00:19:12,860 --> 00:19:14,805
Ready?

389
00:19:14,805 --> 00:19:15,782
AUDIENCE: 12.

390
00:19:15,782 --> 00:19:16,490
PROFESSOR 2: Ooh.

391
00:19:21,780 --> 00:19:24,190
Now what happened?

392
00:19:24,190 --> 00:19:25,060
It's a 12-sided die?

393
00:19:25,060 --> 00:19:27,270
But it could be a
20-sided die, right?

394
00:19:27,270 --> 00:19:29,811
All right, let me just do a few
more, I'll tell you it I get.

395
00:19:29,811 --> 00:19:39,580
I get 1, I get 10, I get 10,
I get 12, I get 3, I get 10.

396
00:19:39,580 --> 00:19:41,060
All right.

397
00:19:41,060 --> 00:19:43,660
Who wants to bet?

398
00:19:43,660 --> 00:19:45,380
What would you be
willing to bet?

399
00:19:45,380 --> 00:19:47,930
Suppose I was willing to
give you like, I don't know,

400
00:19:47,930 --> 00:19:50,700
10 to 1 odds that
it's a 20-sided die.

401
00:19:55,080 --> 00:19:57,640
I mean, of course,
maybe I know something.

402
00:19:57,640 --> 00:19:59,990
I can see the die.

403
00:19:59,990 --> 00:20:01,990
Right, so what we
want to be able to do

404
00:20:01,990 --> 00:20:04,445
is precisely answer
that question.

405
00:20:04,445 --> 00:20:06,570
All right, and it's just
do this Bayesian Updating.

406
00:20:06,570 --> 00:20:09,800
It's not so hard, but organizing
your work is really important.

407
00:20:09,800 --> 00:20:11,000
So, all right.

408
00:20:11,000 --> 00:20:13,190
It was the 12-sided after all.

409
00:20:13,190 --> 00:20:15,180
So you all did good.

410
00:20:15,180 --> 00:20:18,140
All right, so this is
the first board question.

411
00:20:18,140 --> 00:20:18,640
All right.

412
00:20:18,640 --> 00:20:21,700
I want to practice making
the table Jerry showed you.

413
00:20:21,700 --> 00:20:23,640
Make one for if it were 13.

414
00:20:23,640 --> 00:20:26,240
Make one if I had
rolled 5 instead.

415
00:20:26,240 --> 00:20:28,430
Or do the same question
if I had rolled a 9.

416
00:20:28,430 --> 00:20:31,710
In each case, compute the
posterior probabilities

417
00:20:31,710 --> 00:20:33,090
of each of the 5 types of dice.

418
00:20:47,210 --> 00:20:48,960
PROFESSOR: You knew
it was a 20-sided die?

419
00:20:48,960 --> 00:20:49,410
AUDIENCE: Yeah.

420
00:20:49,410 --> 00:20:51,284
PROFESSOR: All right,
what about if it's a 5?

421
00:20:51,284 --> 00:20:52,419
AUDIENCE: Yeah.

422
00:20:52,419 --> 00:20:53,460
PROFESSOR: But that's it.

423
00:20:53,460 --> 00:20:57,580
But you see how that captures
it very nicely in a table.

424
00:20:57,580 --> 00:20:58,540
All of your intuition.

425
00:20:58,540 --> 00:20:59,653
You knew it was a--

426
00:21:26,780 --> 00:21:29,310
PROFESSOR 2: So right now you're
doing the D equals 13 case?

427
00:21:29,310 --> 00:21:31,601
AUDIENCE: Yeah, or I guess
we could combine the tables.

428
00:21:33,395 --> 00:21:34,520
PROFESSOR 2: Oh, I'm sorry.

429
00:21:34,520 --> 00:21:35,400
You're not combining tables.

430
00:21:35,400 --> 00:21:36,590
This is your
likelihood column here?

431
00:21:36,590 --> 00:21:36,930
AUDIENCE: Yeah.

432
00:21:36,930 --> 00:21:38,770
PROFESSOR 2: And then this is
your unnormalized posterior

433
00:21:38,770 --> 00:21:39,550
column.

434
00:21:39,550 --> 00:21:40,590
Is that right?

435
00:21:40,590 --> 00:21:42,640
Or, no this is
actually normalized.

436
00:21:42,640 --> 00:21:44,800
AUDIENCE: This is for if
it were like any number.

437
00:21:44,800 --> 00:21:46,550
Like if we didn't know
what number it was.

438
00:21:48,434 --> 00:21:49,600
PROFESSOR 2: The likelihood?

439
00:21:49,600 --> 00:21:50,050
AUDIENCE: Yes.

440
00:21:50,050 --> 00:21:51,050
PROFESSOR 2: Absolutely.

441
00:21:51,050 --> 00:21:54,931
So the likelihood
doesn't depend on-- wait.

442
00:21:54,931 --> 00:21:55,680
What are we doing?

443
00:21:55,680 --> 00:21:56,900
Rolling a 5 or a 13?

444
00:21:56,900 --> 00:21:58,757
AUDIENCE: Rolling a 13, right?

445
00:21:58,757 --> 00:22:01,090
PROFESSOR 2: Right, so rolling
a 13, for the likelihood,

446
00:22:01,090 --> 00:22:03,510
you're computing the probability
of data that you saw,

447
00:22:03,510 --> 00:22:08,344
the 13, given the hypothesis
say that it's the 4-sided die.

448
00:22:08,344 --> 00:22:09,010
AUDIENCE: Right.

449
00:22:09,010 --> 00:22:10,426
PROFESSOR 2: It's
the 4-sided die,

450
00:22:10,426 --> 00:22:12,480
what's the probability
you would roll a 13?

451
00:22:12,480 --> 00:22:13,115
AUDIENCE: 0.

452
00:22:13,115 --> 00:22:13,906
PROFESSOR 2: Great.

453
00:22:13,906 --> 00:22:14,790
Same with this, this and this.

454
00:22:14,790 --> 00:22:15,850
But what about here?

455
00:22:15,850 --> 00:22:17,527
If it's the 20-sided
die, if that's

456
00:22:17,527 --> 00:22:19,860
your hypothesis, what's the
probability you'd roll a 13.

457
00:22:19,860 --> 00:22:20,840
AUDIENCE: Oh no.

458
00:22:20,840 --> 00:22:21,820
Oh, 1/20.

459
00:22:21,820 --> 00:22:22,800
PROFESSOR 2: Exactly.

460
00:22:22,800 --> 00:22:25,130
So that's the likelihood column.

461
00:22:25,130 --> 00:22:26,992
So this we're going to erase.

462
00:22:26,992 --> 00:22:28,840
OK.

463
00:22:28,840 --> 00:22:31,565
AUDIENCE 2: Then we multiply
this by this and get that.

464
00:22:31,565 --> 00:22:32,440
PROFESSOR 2: Exactly.

465
00:22:32,440 --> 00:22:37,000
AUDIENCE: [INAUDIBLE]
divided by [INAUDIBLE].

466
00:22:37,000 --> 00:22:38,420
PROFESSOR 2: Exactly, exactly.

467
00:22:38,420 --> 00:22:39,757
Does that make sense?

468
00:22:39,757 --> 00:22:41,590
AUDIENCE: Wait, we
multiply these together--

469
00:22:41,590 --> 00:22:43,090
AUDIENCE 2: Multiply
those together.

470
00:22:54,088 --> 00:23:03,474
And then we divide 1 over
100, so it's going to be 0--

471
00:23:03,474 --> 00:23:04,390
PROFESSOR 2: So, yeah.

472
00:23:04,390 --> 00:23:06,740
So you get to just--

473
00:23:06,740 --> 00:23:07,657
AUDIENCE: And then HD.

474
00:23:07,657 --> 00:23:08,489
PROFESSOR 2: No, no.

475
00:23:08,489 --> 00:23:09,610
Oh yeah, you're right.

476
00:23:09,610 --> 00:23:10,110
HD.

477
00:23:12,630 --> 00:23:17,350
One thing I want to point out
is instead of using a comma,

478
00:23:17,350 --> 00:23:19,810
all right, this is
conditional probability

479
00:23:19,810 --> 00:23:24,070
so it's probability of
D given H, like this.

480
00:23:24,070 --> 00:23:24,660
OK.

481
00:23:24,660 --> 00:23:26,455
There you go.

482
00:23:26,455 --> 00:23:27,580
So that's absolutely right.

483
00:23:27,580 --> 00:23:29,630
At the end, we want to
know how likely each

484
00:23:29,630 --> 00:23:31,000
die was given the data.

485
00:23:31,000 --> 00:23:33,362
AUDIENCE: Because
this total is just P.

486
00:23:33,362 --> 00:23:34,445
PROFESSOR 2: That's right.

487
00:23:37,790 --> 00:23:47,700
Which is 1 over 100.

488
00:23:47,700 --> 00:23:48,200
Great.

489
00:23:48,200 --> 00:23:49,587
So now, do it again.

490
00:23:49,587 --> 00:23:50,170
Yeah, exactly.

491
00:23:50,170 --> 00:23:51,836
You don't even need
to make a new table.

492
00:24:02,450 --> 00:24:04,692
PROFESSOR 2: Awesome.

493
00:24:04,692 --> 00:24:06,604
AUDIENCE: So this
column [INAUDIBLE]

494
00:24:06,604 --> 00:24:08,516
to the probability,
because [INAUDIBLE].

495
00:24:11,980 --> 00:24:14,692
PROFESSOR 2: Yeah, you take each
hypothesis times probability

496
00:24:14,692 --> 00:24:16,150
of data given the
hypothesis, which

497
00:24:16,150 --> 00:24:20,210
is the same as the
probability of D and H.

498
00:24:20,210 --> 00:24:21,210
AUDIENCE: Yeah, exactly.

499
00:24:21,210 --> 00:24:23,170
PROFESSOR 2: So your
partitioning it up, sort of,

500
00:24:23,170 --> 00:24:24,170
based on the hypotheses.

501
00:24:24,170 --> 00:24:27,502
AUDIENCE: Because this equals
that times probability of D.

502
00:24:27,502 --> 00:24:28,585
PROFESSOR 2: That's right.

503
00:24:28,585 --> 00:24:29,724
AUDIENCE: D confuses me.

504
00:24:29,724 --> 00:24:30,500
What is P of D?

505
00:24:30,500 --> 00:24:32,041
PROFESSOR 2:
Probability of the data.

506
00:24:32,041 --> 00:24:33,770
So, right.

507
00:24:33,770 --> 00:24:37,020
So a priori, you don't know
which die it is, right?

508
00:24:37,020 --> 00:24:40,810
But there's still a
probability of getting a 13.

509
00:24:40,810 --> 00:24:41,420
Right?

510
00:24:41,420 --> 00:24:43,140
And what is the probability?

511
00:24:43,140 --> 00:24:46,140
Well first, we would
have to, in this case,

512
00:24:46,140 --> 00:24:47,685
we could only pick
the 20-sided die,

513
00:24:47,685 --> 00:24:49,470
that has a 1/5
chance of happening.

514
00:24:49,470 --> 00:24:51,970
And then, given that we
picked the 20-sided die,

515
00:24:51,970 --> 00:24:55,010
we have to roll a 13,
which is a 1/20 chance.

516
00:24:55,010 --> 00:24:59,736
Which is exactly why
this sums to 1 in a 100.

517
00:24:59,736 --> 00:25:01,110
And then more
generally, you have

518
00:25:01,110 --> 00:25:02,318
the Law of Total Probability.

519
00:25:09,310 --> 00:25:15,875
So here, I think you should
keep in mind which is the D

520
00:25:15,875 --> 00:25:18,230
and which is H.
Maybe write it above,

521
00:25:18,230 --> 00:25:24,050
otherwise it's a little
confusing to know what-- great.

522
00:25:24,050 --> 00:25:26,610
OK, and so these are
the final probabilities

523
00:25:26,610 --> 00:25:29,138
you got if you rolled a 9.

524
00:25:29,138 --> 00:25:30,620
Is that right?

525
00:25:30,620 --> 00:25:32,775
So there's something
off here right?

526
00:25:32,775 --> 00:25:34,358
AUDIENCE: Sorry,
these are our priors.

527
00:25:36,464 --> 00:25:38,750
I'm not labeling this.

528
00:25:38,750 --> 00:25:39,458
These are priors.

529
00:25:39,458 --> 00:25:40,955
This is gonna be [INAUDIBLE].

530
00:25:45,132 --> 00:25:47,590
PROFESSOR 2: So actually here,
I think you have likelihood.

531
00:25:47,590 --> 00:25:50,230
This is likelihood
here, exactly.

532
00:25:50,230 --> 00:25:52,390
That's likelihood.

533
00:25:52,390 --> 00:25:54,710
Your prior should be what?

534
00:25:54,710 --> 00:25:55,949
AUDIENCE: 0.2.

535
00:25:55,949 --> 00:25:57,240
PROFESSOR 2: What should it be?

536
00:25:57,240 --> 00:25:58,030
AUDIENCE: 0.2 for each one.

537
00:25:58,030 --> 00:25:58,410
PROFESSOR 2: Yeah, 1/5.

538
00:25:58,410 --> 00:25:59,576
Oh, I didn't see the points.

539
00:25:59,576 --> 00:26:02,260
Yeah, so that's your
prior, 1/5 for each.

540
00:26:02,260 --> 00:26:03,620
This is your likelihood.

541
00:26:03,620 --> 00:26:05,304
Then you multiply
them, and that's

542
00:26:05,304 --> 00:26:06,720
the unnormalized
posterior, right?

543
00:26:09,426 --> 00:26:11,050
Let's have a brief
discussion about it,

544
00:26:11,050 --> 00:26:11,980
and then we're going
to give you something

545
00:26:11,980 --> 00:26:13,063
a little more challenging.

546
00:26:15,910 --> 00:26:20,050
So for rolling a 13, right,
the first thing you do

547
00:26:20,050 --> 00:26:22,510
is ask, what is the prior?

548
00:26:22,510 --> 00:26:23,980
And you saw, I had 5 dice.

549
00:26:23,980 --> 00:26:25,370
I shook it around.

550
00:26:25,370 --> 00:26:27,840
A student reached in
randomly, not even me.

551
00:26:27,840 --> 00:26:31,800
So we can trust that it
was 1/5 for each die.

552
00:26:31,800 --> 00:26:35,330
Equally likely, that's a
uniform prior, right, discrete

553
00:26:35,330 --> 00:26:37,920
probability mass function.

554
00:26:37,920 --> 00:26:39,200
Great.

555
00:26:39,200 --> 00:26:40,140
Likelihood.

556
00:26:40,140 --> 00:26:42,430
Well, that's going to
depend on the data,

557
00:26:42,430 --> 00:26:45,740
because it's the probability of
the data given the hypotheses.

558
00:26:45,740 --> 00:26:49,310
The hypothesis in this case
is how many sides on the die.

559
00:26:49,310 --> 00:26:53,430
So, if the data's 13, what's
the probability that I

560
00:26:53,430 --> 00:26:57,950
would roll that data, a 13,
given that it was the 4-sided

561
00:26:57,950 --> 00:26:59,300
die.

562
00:26:59,300 --> 00:26:59,890
0.

563
00:26:59,890 --> 00:27:00,390
Right?

564
00:27:00,390 --> 00:27:01,150
It's impossible.

565
00:27:01,150 --> 00:27:04,780
Similarly 4, 6, 8 and 12 sides.

566
00:27:04,780 --> 00:27:07,440
But given that I had
the 20-sided die,

567
00:27:07,440 --> 00:27:10,850
there would be a 1/20
probability of rolling a 13.

568
00:27:10,850 --> 00:27:12,306
That's the likelihood column.

569
00:27:12,306 --> 00:27:13,280
All right.

570
00:27:13,280 --> 00:27:16,140
The next column, what we called
the unnormalized posterior, we

571
00:27:16,140 --> 00:27:19,840
multiply, everything is 0,
and then we get 1 out of 100

572
00:27:19,840 --> 00:27:25,140
as the product for
the 20-sided die.

573
00:27:25,140 --> 00:27:29,100
If we sum this up by the
Law of Total Probability,

574
00:27:29,100 --> 00:27:31,201
we get the probability
of the data.

575
00:27:31,201 --> 00:27:32,950
Now, a student asked,
what does that mean?

576
00:27:32,950 --> 00:27:34,770
We don't know which die it was?

577
00:27:34,770 --> 00:27:37,850
In this case, what
it means is that it's

578
00:27:37,850 --> 00:27:42,630
reasonable to ask the question,
well, if someone reaches in,

579
00:27:42,630 --> 00:27:45,780
grabs 1 of these 5 dice
randomly and rolls it

580
00:27:45,780 --> 00:27:48,070
and gets-- I mean,
what's the probability

581
00:27:48,070 --> 00:27:50,440
that the result would be a 13.

582
00:27:50,440 --> 00:27:50,980
Right?

583
00:27:50,980 --> 00:27:53,270
That's a reasonable question.

584
00:27:53,270 --> 00:27:56,790
And in this case, it's not so
hard to analyze that directly,

585
00:27:56,790 --> 00:27:58,870
because the only way
that could happen

586
00:27:58,870 --> 00:28:02,780
is if they picked the 20-sided
die and then rolled a 13.

587
00:28:02,780 --> 00:28:06,370
Well, given that they
picked the 20-sided die.

588
00:28:06,370 --> 00:28:09,440
Well, there's a 1/5 chance
they picked the 20-sided die,

589
00:28:09,440 --> 00:28:13,010
and then a 1/20 chance they
roll of 13 given that they

590
00:28:13,010 --> 00:28:15,080
picked the 20-sided die.

591
00:28:15,080 --> 00:28:20,880
So the probability of rolling
a 13 overall is 1 in 100.

592
00:28:20,880 --> 00:28:23,369
The probability of the data.

593
00:28:23,369 --> 00:28:25,160
PROFESSOR: John, let
me just say one thing.

594
00:28:25,160 --> 00:28:27,035
The way we would have
done this before is you

595
00:28:27,035 --> 00:28:28,990
would have made a tree.

596
00:28:28,990 --> 00:28:30,690
The top branch of
the tree would have

597
00:28:30,690 --> 00:28:33,440
gone to the 5 types of die,
and then each of those die

598
00:28:33,440 --> 00:28:35,570
would have gone to
all the possibilities.

599
00:28:35,570 --> 00:28:39,340
And the only branch
that leads to a 13

600
00:28:39,340 --> 00:28:42,874
is down to the 20-sided die
and then down to the 13.

601
00:28:42,874 --> 00:28:44,540
PROFESSOR 2: Notice
also that this would

602
00:28:44,540 --> 00:28:47,100
be a very big tree, right?

603
00:28:47,100 --> 00:28:50,700
You have 5 branches, and
then this would have 4 and 6,

604
00:28:50,700 --> 00:28:53,480
and 8, and 12, and
20 coming off of it.

605
00:28:53,480 --> 00:28:54,630
Finally we normalize.

606
00:28:54,630 --> 00:28:57,330
Using Bayes Rule, this gives
us a posterior probability mass

607
00:28:57,330 --> 00:28:57,890
function.

608
00:28:57,890 --> 00:29:01,370
It's 0 for everything but
the 20-sided hypothesis.

609
00:29:01,370 --> 00:29:06,090
Therefore, we get 100%
belief, 100% probability,

610
00:29:06,090 --> 00:29:09,930
given that we rolled a 13,
that it was the 20-sided die.

611
00:29:09,930 --> 00:29:12,870
Now of course you all
knew that immediately.

612
00:29:12,870 --> 00:29:15,140
If you roll a 13, it has
to be the 20-sided die.

613
00:29:15,140 --> 00:29:17,540
So let's look at a more
interesting one that you did,

614
00:29:17,540 --> 00:29:18,050
the 5.

615
00:29:18,050 --> 00:29:18,950
So what changes?

616
00:29:18,950 --> 00:29:21,270
That's the big question.

617
00:29:21,270 --> 00:29:27,494
The only thing that
changes is the likelihood.

618
00:29:27,494 --> 00:29:28,910
I mean, granted,
the rest changes.

619
00:29:28,910 --> 00:29:30,540
But everything in
these two columns

620
00:29:30,540 --> 00:29:32,250
are computed from the first two.

621
00:29:32,250 --> 00:29:37,080
The prior stays the same, but
the likelihood function changes

622
00:29:37,080 --> 00:29:38,860
because we have different data.

623
00:29:38,860 --> 00:29:40,890
In this case, if we
roll a 5, we can't

624
00:29:40,890 --> 00:29:42,980
get that on the 4-sided die.

625
00:29:42,980 --> 00:29:45,930
We have a 1/6 chance on the
6-sided, 1/8 on the 8-sided,

626
00:29:45,930 --> 00:29:48,970
1/12 on the 12-sided and
1/20 on the 20-sided.

627
00:29:48,970 --> 00:29:52,270
Multiply those columns, get
the unnormalized probability,

628
00:29:52,270 --> 00:29:54,860
or the unnormalized posterior.

629
00:29:54,860 --> 00:29:58,040
That sums to the
probability of getting a 5.

630
00:29:58,040 --> 00:30:01,160
When you divide by it, it
normalizes the posterior,

631
00:30:01,160 --> 00:30:02,990
and what do we see?

632
00:30:02,990 --> 00:30:07,260
Well, of course you couldn't
have used the 4-sided die,

633
00:30:07,260 --> 00:30:11,500
and of the remaining the most
likely, the most probable,

634
00:30:11,500 --> 00:30:15,560
given the data, is
the 6-sided die.

635
00:30:15,560 --> 00:30:17,310
It has a probability,
that hypothesis

636
00:30:17,310 --> 00:30:21,470
has a probability of about 40%,
compared to the 20-sided die,

637
00:30:21,470 --> 00:30:24,020
which is only about 12%.

638
00:30:24,020 --> 00:30:25,270
OK.

639
00:30:25,270 --> 00:30:27,450
Last one.

640
00:30:27,450 --> 00:30:28,620
So this is the same deal.

641
00:30:28,620 --> 00:30:31,530
Again, the likelihood
function changes.

642
00:30:31,530 --> 00:30:32,890
The prior's the same.

643
00:30:32,890 --> 00:30:34,230
We go through the table.

644
00:30:34,230 --> 00:30:36,450
The only two possibilities
are the 12 and 20-sided,

645
00:30:36,450 --> 00:30:38,850
and the 12-sided, given
that you rolled a 9,

646
00:30:38,850 --> 00:30:42,360
is about twice as likely
as the 20-sided, which

647
00:30:42,360 --> 00:30:45,680
should fit with your intuition.

648
00:30:45,680 --> 00:30:46,530
OK.

649
00:30:46,530 --> 00:30:50,150
So next, we want you to
not erase, but go back

650
00:30:50,150 --> 00:30:51,810
to the boards
you've already made,

651
00:30:51,810 --> 00:30:54,370
and we're going to explore a
little bit the idea of repeated

652
00:30:54,370 --> 00:30:55,070
trials.

653
00:30:55,070 --> 00:30:55,570
Right?

654
00:30:55,570 --> 00:30:58,070
Often if you're collecting
data, you don't just

655
00:30:58,070 --> 00:31:00,120
collect one data
point, you collect

656
00:31:00,120 --> 00:31:02,860
a series of data points,
maybe a series of patients

657
00:31:02,860 --> 00:31:03,950
in a clinical trial.

658
00:31:03,950 --> 00:31:05,520
You have data coming in, right?

659
00:31:05,520 --> 00:31:06,730
And you can update.

660
00:31:06,730 --> 00:31:09,110
You can update each
time you get more data.

661
00:31:09,110 --> 00:31:12,585
And when you do this, your
prior becomes a posterior

662
00:31:12,585 --> 00:31:13,960
in the first
update, and then you

663
00:31:13,960 --> 00:31:17,110
use that posterior as the
prior for the next update.

664
00:31:17,110 --> 00:31:21,680
Your beliefs are being
updated as more data comes in.

665
00:31:21,680 --> 00:31:24,390
If you think about it, like when
you walk outside this course,

666
00:31:24,390 --> 00:31:26,694
if you're like me
you'll sort of suddenly

667
00:31:26,694 --> 00:31:28,610
realize you're doing
this process all the time

668
00:31:28,610 --> 00:31:30,500
in your life, and you'll start
approaching problems this way.

669
00:31:30,500 --> 00:31:31,874
And if you really
like me, you'll

670
00:31:31,874 --> 00:31:34,530
change your religion on
Facebook to Bayesian.

671
00:31:34,530 --> 00:31:37,110
But you don't have
to be that extreme.

672
00:31:37,110 --> 00:31:39,180
So great.

673
00:31:39,180 --> 00:31:42,970
So here what we want you to do
is pretend that I roll the die

674
00:31:42,970 --> 00:31:43,860
and then got a 5.

675
00:31:43,860 --> 00:31:45,320
And then I rolled the same die.

676
00:31:45,320 --> 00:31:46,200
I didn't get a different one.

677
00:31:46,200 --> 00:31:47,824
I just rolled the
same die, just like I

678
00:31:47,824 --> 00:31:49,950
did in that demonstration,
and then I got a 9.

679
00:31:49,950 --> 00:31:54,110
OK, so now we have a sequence
of two pieces of data, 5 and 9.

680
00:31:54,110 --> 00:32:06,120
So I want to find the posterior
that-- update in two steps.

681
00:32:06,120 --> 00:32:07,590
I'm doing this one.

682
00:32:07,590 --> 00:32:08,990
I'm doing this one!

683
00:32:08,990 --> 00:32:13,895
OK, so let me show you how to
do this, and then you'll do it.

684
00:32:13,895 --> 00:32:15,770
OK, so first we're going
to update for the 5.

685
00:32:15,770 --> 00:32:19,310
Then we're going to
update again for the 9.

686
00:32:19,310 --> 00:32:21,530
Magic!

687
00:32:21,530 --> 00:32:24,410
The prior, 1/5 for each, right?

688
00:32:24,410 --> 00:32:26,030
That's where we're
starting from.

689
00:32:26,030 --> 00:32:27,400
We roll a 5.

690
00:32:27,400 --> 00:32:28,890
That's our likelihood.

691
00:32:28,890 --> 00:32:31,410
Again, you get a
0 for the 4-sided

692
00:32:31,410 --> 00:32:35,070
and you get 1 over n for
each of the other n-sided

693
00:32:35,070 --> 00:32:36,050
possibilities.

694
00:32:36,050 --> 00:32:37,300
We multiply.

695
00:32:37,300 --> 00:32:39,090
That gives us our
unnormalized posterior

696
00:32:39,090 --> 00:32:40,630
for the first piece of data.

697
00:32:40,630 --> 00:32:41,420
Great.

698
00:32:41,420 --> 00:32:45,620
Now that is going
to be our new prior.

699
00:32:45,620 --> 00:32:47,570
Now, you may complain,
wait a minute.

700
00:32:47,570 --> 00:32:49,170
This doesn't add to 1.

701
00:32:49,170 --> 00:32:53,340
But it's OK, because we're
going to do it again.

702
00:32:53,340 --> 00:32:56,430
At the very end we'll normalize,
and we'll get the same answer

703
00:32:56,430 --> 00:32:59,240
as if we had normalized here
and then normalized again.

704
00:32:59,240 --> 00:33:01,840
Because we're just
multiplying the whole column

705
00:33:01,840 --> 00:33:03,700
by the same scalar, by
the same real number.

706
00:33:03,700 --> 00:33:04,200
OK?

707
00:33:04,200 --> 00:33:06,020
So it saves you
work to not bother

708
00:33:06,020 --> 00:33:08,951
to normalize the first time.

709
00:33:08,951 --> 00:33:09,450
Great.

710
00:33:09,450 --> 00:33:11,440
So that's our new prior.

711
00:33:11,440 --> 00:33:13,740
We get our next
piece of data, the 9.

712
00:33:13,740 --> 00:33:18,460
We have 0, 0, 0, 1/12, and
1/20 as the likelihoods.

713
00:33:18,460 --> 00:33:22,450
We multiply this new
prior by this likelihood,

714
00:33:22,450 --> 00:33:25,220
get the new
unnormalized posterior.

715
00:33:25,220 --> 00:33:30,390
It only has two possible
nonzero probabilities in it.

716
00:33:30,390 --> 00:33:35,680
The sum, the total probability
of rolling a 5 and then a 9

717
00:33:35,680 --> 00:33:44,646
is only 0.0019, OK?

718
00:33:44,646 --> 00:33:47,001
PROFESSOR: That's [INAUDIBLE]
unnormalized [INAUDIBLE].

719
00:33:47,001 --> 00:33:48,890
PROFESSOR 2: Ah!

720
00:33:48,890 --> 00:33:49,580
Jerry caught me.

721
00:33:49,580 --> 00:33:51,730
I was hoping a student
would catch me.

722
00:33:51,730 --> 00:33:52,470
OK, maybe not.

723
00:33:52,470 --> 00:33:54,550
Maybe I was fooling myself.

724
00:33:54,550 --> 00:33:56,130
We haven't normalized.

725
00:33:56,130 --> 00:33:59,250
So, it's not quite right.

726
00:33:59,250 --> 00:34:01,800
If we had normalized here, then
by the time we got to here,

727
00:34:01,800 --> 00:34:05,280
we could interpret this
as that total probability.

728
00:34:05,280 --> 00:34:07,890
But in any case, the
point is we can normalize

729
00:34:07,890 --> 00:34:10,750
to get to the final stage,
this final posterior,

730
00:34:10,750 --> 00:34:13,120
by dividing by the sum here.

731
00:34:13,120 --> 00:34:15,527
Now it adds to 1, and
at the end of the day,

732
00:34:15,527 --> 00:34:16,360
what do we conclude?

733
00:34:16,360 --> 00:34:20,199
We conclude that it's almost
three times more likely

734
00:34:20,199 --> 00:34:23,770
that it was the 12-sided
die than that it

735
00:34:23,770 --> 00:34:26,100
was the 20-sided die.

736
00:34:26,100 --> 00:34:26,600
Right?

737
00:34:26,600 --> 00:34:29,880
So this is very similar to
what happened when randomly

738
00:34:29,880 --> 00:34:31,600
had the 12-sided and rolled it.

739
00:34:31,600 --> 00:34:34,800
After two times, I had data
that looked a lot like this,

740
00:34:34,800 --> 00:34:37,290
and if we had rigorously
computed those probabilities,

741
00:34:37,290 --> 00:34:39,170
this is what we
would have gotten.

742
00:34:39,170 --> 00:34:41,489
OK?

743
00:34:41,489 --> 00:34:44,639
So now, now it's your turn.

744
00:34:44,639 --> 00:34:48,190
So, we want to try the same
thing, but in the other order.

745
00:34:48,190 --> 00:34:51,100
Suppose I first roll the
9 and then I roll the 5.

746
00:34:51,100 --> 00:34:52,469
So that's the first question.

747
00:34:52,469 --> 00:34:55,600
The second question is,
can you do it in one go?

748
00:34:55,600 --> 00:34:59,120
Instead of doing it updating
for the 9 and then for the 5,

749
00:34:59,120 --> 00:35:02,340
can you incorporate
both the 9 and the 5

750
00:35:02,340 --> 00:35:05,800
into a single
likelihood function

751
00:35:05,800 --> 00:35:07,520
and do the updating in one step?

752
00:35:07,520 --> 00:35:10,080
All right, so try these and
think about what's similar,

753
00:35:10,080 --> 00:35:12,510
what's different
than what I just did.

754
00:35:12,510 --> 00:35:13,920
You don't have to.

755
00:35:13,920 --> 00:35:15,747
You don't have to.

756
00:35:15,747 --> 00:35:17,330
You could normalize
it, then you would

757
00:35:17,330 --> 00:35:19,000
multiply by the likelihood.

758
00:35:19,000 --> 00:35:20,850
You'd get something
again unnormalized,

759
00:35:20,850 --> 00:35:22,544
and then you'd have
normalize again,

760
00:35:22,544 --> 00:35:24,210
and then you'd get
to your final answer.

761
00:35:24,210 --> 00:35:27,430
Or, don't bother
normalizing at this stage.

762
00:35:27,430 --> 00:35:30,290
Just multiply this by the
likelihood, and then just only

763
00:35:30,290 --> 00:35:31,600
normalize once at the very end.

764
00:35:31,600 --> 00:35:34,580
And you'll get the same
answer because, you know,

765
00:35:34,580 --> 00:35:36,530
normalizing once and
then normalizing again

766
00:35:36,530 --> 00:35:39,704
is the same as just
normalizing at the very end.

767
00:35:39,704 --> 00:35:41,620
You're just multiplying
a vector, this column,

768
00:35:41,620 --> 00:35:44,585
by rescaling it to
make it unit length--

769
00:35:44,585 --> 00:35:46,251
AUDIENCE: And then
you're multiplying it

770
00:35:46,251 --> 00:35:47,043
by something else--

771
00:35:47,043 --> 00:35:47,917
PROFESSOR 2: Exactly.

772
00:35:47,917 --> 00:35:48,890
Exactly.

773
00:35:48,890 --> 00:35:50,616
AUDIENCE: Isn't
number 1 going to be

774
00:35:50,616 --> 00:35:52,945
the same answer as the
example we just did,

775
00:35:52,945 --> 00:35:54,820
because you're going to
be updating-- I mean,

776
00:35:54,820 --> 00:35:55,694
it's going different.

777
00:35:55,694 --> 00:35:58,061
OK, we just have to do it.

778
00:35:58,061 --> 00:35:59,060
PROFESSOR 2: Absolutely.

779
00:35:59,060 --> 00:36:01,699
So, yes, absolutely.

780
00:36:01,699 --> 00:36:03,490
All of these could give
you the same answer

781
00:36:03,490 --> 00:36:05,156
because it's the same
exact data, right?

782
00:36:05,156 --> 00:36:06,790
Starting from the same priors.

783
00:36:06,790 --> 00:36:08,400
But what we want
you to really see

784
00:36:08,400 --> 00:36:10,702
is how the math proves that.

785
00:36:10,702 --> 00:36:13,160
AUDIENCE: I mean we're just
multiplying in different order.

786
00:36:13,160 --> 00:36:14,243
PROFESSOR 2: That's right.

787
00:36:14,243 --> 00:36:16,900
So it all comes down to what?

788
00:36:16,900 --> 00:36:19,280
What property of multiplication?

789
00:36:19,280 --> 00:36:20,944
AUDIENCE: Associative?

790
00:36:20,944 --> 00:36:21,860
AUDIENCE: Commutative.

791
00:36:21,860 --> 00:36:22,820
PROFESSOR 2: Commutative.

792
00:36:22,820 --> 00:36:24,486
I guess you're also
using associativity.

793
00:36:24,486 --> 00:36:27,290
It's just that multiplication
is commutative, right?

794
00:36:27,290 --> 00:36:30,640
So it doesn't matter what order
you multiply these columns in,

795
00:36:30,640 --> 00:36:32,776
the final result's
going to be the same.

796
00:36:32,776 --> 00:36:35,860
So just about how the
order changes depending

797
00:36:35,860 --> 00:36:38,740
on the order of the 5
versus the 9, and also,

798
00:36:38,740 --> 00:36:40,490
what happens if you
think of both at once,

799
00:36:40,490 --> 00:36:42,837
how would you then make a
table directly for that?

800
00:36:42,837 --> 00:36:43,670
[INTERPOSING VOICES]

801
00:37:03,650 --> 00:37:04,220
PROFESSOR: 5.

802
00:37:04,220 --> 00:37:04,720
Oh, I see.

803
00:37:04,720 --> 00:37:08,602
So you multiplied-- here,
is this your likelihood?

804
00:37:08,602 --> 00:37:09,560
So you multiplied that.

805
00:37:09,560 --> 00:37:11,890
Where's your plus times this?

806
00:37:11,890 --> 00:37:14,850
Oh, you even normalized.

807
00:37:14,850 --> 00:37:15,357
Right.

808
00:37:15,357 --> 00:37:16,940
So did you understand
John's statement

809
00:37:16,940 --> 00:37:19,016
that you didn't actually
have to normalize?

810
00:37:19,016 --> 00:37:19,750
AUDIENCE: Right.

811
00:37:19,750 --> 00:37:20,960
I mean, if you'd
already done it.

812
00:37:20,960 --> 00:37:22,126
PROFESSOR: No, I understand.

813
00:37:22,126 --> 00:37:23,430
It's a nicer number than 1/60.

814
00:37:23,430 --> 00:37:24,860
But if you didn't
have it, you could

815
00:37:24,860 --> 00:37:27,068
have just used this column
and normalized at the end.

816
00:37:27,068 --> 00:37:27,867
That make sense?

817
00:37:27,867 --> 00:37:28,367
Excellent.

818
00:37:35,049 --> 00:37:35,840
I'm not trying to--

819
00:37:35,840 --> 00:37:37,820
AUDIENCE: This is just
if you do it in one step.

820
00:37:37,820 --> 00:37:38,286
PROFESSOR: I see.

821
00:37:38,286 --> 00:37:39,000
AUDIENCE: You just multiply.

822
00:37:39,000 --> 00:37:40,040
PROFESSOR: Did you
get different answers?

823
00:37:40,040 --> 00:37:40,740
AUDIENCE: No, it's the same.

824
00:37:40,740 --> 00:37:41,250
PROFESSOR: It's the same.

825
00:37:41,250 --> 00:37:41,920
It makes sense.

826
00:37:41,920 --> 00:37:42,670
You have the data.

827
00:37:42,670 --> 00:37:44,852
That's the evidence you have.

828
00:37:44,852 --> 00:37:46,620
Right.

829
00:37:46,620 --> 00:37:48,342
So was this all clear?

830
00:37:48,342 --> 00:37:49,800
Was John's statement
about not only

831
00:37:49,800 --> 00:37:51,440
needing the
unnormalized posterior

832
00:37:51,440 --> 00:37:54,310
in the middle reasonably clear?

833
00:37:54,310 --> 00:37:55,810
AUDIENCE: What were
talking about is

834
00:37:55,810 --> 00:37:59,394
if you were to use
the normalized one,

835
00:37:59,394 --> 00:38:01,386
would then your
unnormalized posterior

836
00:38:01,386 --> 00:38:03,690
be the actual probabilities?

837
00:38:03,690 --> 00:38:04,860
PROFESSOR: No.

838
00:38:04,860 --> 00:38:09,360
No, because in order to use
the Law of Total Probability,

839
00:38:09,360 --> 00:38:10,860
you have to multiply
your likelihood

840
00:38:10,860 --> 00:38:12,730
by genuine probabilities.

841
00:38:12,730 --> 00:38:15,407
And the unnormalized posterior
is not a genuine probability.

842
00:38:15,407 --> 00:38:17,240
AUDIENCE: But if you
use the normalized one.

843
00:38:17,240 --> 00:38:18,990
PROFESSOR: If you used
the normalized one,

844
00:38:18,990 --> 00:38:24,210
then it would be-- that would be
the total probability of that.

845
00:38:24,210 --> 00:38:25,235
That's exactly right.

846
00:38:25,235 --> 00:38:26,610
AUDIENCE: But, in
this case, it's

847
00:38:26,610 --> 00:38:28,050
this that's the total
probability, right?

848
00:38:28,050 --> 00:38:28,716
PROFESSOR: Yeah.

849
00:38:28,716 --> 00:38:31,440
This is now your
posterior probability.

850
00:38:31,440 --> 00:38:32,690
These are now the probability.

851
00:38:32,690 --> 00:38:34,110
So, make a distinction.

852
00:38:34,110 --> 00:38:36,970
The total probability
is about the data,

853
00:38:36,970 --> 00:38:42,249
and these probabilities here
are about the hypotheses.

854
00:38:42,249 --> 00:38:45,182
AUDIENCE: Sort of like the fit
of the hypothesis to the data.

855
00:38:45,182 --> 00:38:46,640
PROFESSOR: You
could think of that.

856
00:38:46,640 --> 00:38:47,890
They're genuine probabilities.

857
00:38:47,890 --> 00:38:52,700
This is the probability that
you chose the 12-sided die given

858
00:38:52,700 --> 00:38:53,200
the data.

859
00:38:53,200 --> 00:38:59,410
That is, if I did this a
million times and every time,

860
00:38:59,410 --> 00:39:02,360
I only looked at the times
I rolled a 9 and then the 5,

861
00:39:02,360 --> 00:39:05,160
you would find this
fraction of those times

862
00:39:05,160 --> 00:39:07,460
would be-- it
would be a 12-sided

863
00:39:07,460 --> 00:39:09,060
and this fraction
would be a 20-sided.

864
00:39:09,060 --> 00:39:12,320
So, I forget what the
numbers were, like 3 to 1.

865
00:39:12,320 --> 00:39:12,920
Yeah.

866
00:39:12,920 --> 00:39:13,942
Makes sense?

867
00:39:13,942 --> 00:39:15,270
Excellent.

868
00:39:15,270 --> 00:39:19,010
AUDIENCE: So we multiply
them together, but then--

869
00:39:19,010 --> 00:39:21,170
PROFESSOR 2: You get the
same exact answer, right?

870
00:39:21,170 --> 00:39:23,332
AUDIENCE: So we get
these, this answer.

871
00:39:23,332 --> 00:39:25,790
AUDIENCE 2: That's right,
so it's unnormalized.

872
00:39:25,790 --> 00:39:26,520
PROFESSOR: And
then you normalize

873
00:39:26,520 --> 00:39:27,770
it and you get this, right.

874
00:39:27,770 --> 00:39:31,015
AUDIENCE: There's no way
to normalize it without--

875
00:39:31,015 --> 00:39:33,390
PROFESSOR 2: All right, let's
come back together and take

876
00:39:33,390 --> 00:39:35,070
a look at what happened here.

877
00:39:38,470 --> 00:39:39,910
I saw some aha moments.

878
00:39:39,910 --> 00:39:40,590
That was great.

879
00:39:43,410 --> 00:39:45,160
Some people cited The
Commutative Property

880
00:39:45,160 --> 00:39:46,040
of Multiplication.

881
00:39:46,040 --> 00:39:47,040
That made me very happy.

882
00:39:49,510 --> 00:39:52,100
So, what do we do?

883
00:39:52,100 --> 00:39:54,175
So when we do the
9 and then the 5,

884
00:39:54,175 --> 00:39:56,300
it looks just like what I
did with the 5 and the 9,

885
00:39:56,300 --> 00:39:59,240
except the likelihood
columns are flipped.

886
00:39:59,240 --> 00:40:04,490
So we have our prior,
the likelihood for the 9

887
00:40:04,490 --> 00:40:07,050
gives us this column.

888
00:40:07,050 --> 00:40:10,550
The product gives us an,
first, unnormalized posterior.

889
00:40:10,550 --> 00:40:14,540
Then we do the likelihood
for rolling a 5.

890
00:40:14,540 --> 00:40:17,000
The product of the
unnormalized posterior

891
00:40:17,000 --> 00:40:18,750
and the second
likelihood gives us

892
00:40:18,750 --> 00:40:22,410
our final unnormalized
posterior.

893
00:40:22,410 --> 00:40:25,550
And notice the sum is still
exactly what it was before,

894
00:40:25,550 --> 00:40:27,910
0.019.

895
00:40:27,910 --> 00:40:30,940
When we normalize, we get
exactly the same answer

896
00:40:30,940 --> 00:40:31,810
that we had before.

897
00:40:31,810 --> 00:40:34,720
Now, if you contrast
this with the table

898
00:40:34,720 --> 00:40:36,680
we looked at when we
did the 5 and then

899
00:40:36,680 --> 00:40:41,940
the 9, what changed is these
two columns were flipped around.

900
00:40:41,940 --> 00:40:44,210
And so I suppose this
column was different

901
00:40:44,210 --> 00:40:45,960
and this column was
different as a result.

902
00:40:45,960 --> 00:40:50,740
And then the final result,
however, was exactly the same.

903
00:40:50,740 --> 00:40:53,674
Now, what if we do
them both at once?

904
00:40:53,674 --> 00:40:54,590
What should that mean?

905
00:40:57,340 --> 00:40:59,500
If we do them both
at once, then,

906
00:40:59,500 --> 00:41:02,220
well we start from the
same uniform prior,

907
00:41:02,220 --> 00:41:05,780
and our likelihood-- now,
we should think of the data

908
00:41:05,780 --> 00:41:09,700
as both rolls, the 5 and the 9.

909
00:41:09,700 --> 00:41:10,790
OK.

910
00:41:10,790 --> 00:41:15,100
Now what's important here
is that if we condition

911
00:41:15,100 --> 00:41:16,400
on a given hypothesis.

912
00:41:16,400 --> 00:41:18,970
For example, if we
condition on the hypothesis

913
00:41:18,970 --> 00:41:22,980
that it's a 12-sided
die, then the first roll

914
00:41:22,980 --> 00:41:27,130
and the second roll are
independent events, right?

915
00:41:27,130 --> 00:41:31,260
It means that we can figure
out the probability of a 5

916
00:41:31,260 --> 00:41:33,430
and then a 9 on
the 12-sided die.

917
00:41:33,430 --> 00:41:35,610
It's just the product
of the probability

918
00:41:35,610 --> 00:41:38,820
of a 5 and a 12-sided die
and the probability of a 9

919
00:41:38,820 --> 00:41:40,140
on a 12-sided die.

920
00:41:40,140 --> 00:41:42,270
That's why we're justified
in taking the product.

921
00:41:42,270 --> 00:41:45,890
If we condition on the die
that it is, on the hypothesis,

922
00:41:45,890 --> 00:41:50,280
12-sided, then the rolls are
all independent of each other.

923
00:41:50,280 --> 00:41:55,070
In particular, we get 1/12
times 1/12 for rolling a 5

924
00:41:55,070 --> 00:41:56,080
and then rolling a 9.

925
00:41:56,080 --> 00:41:58,820
And for the 20-sided die,
we get 1/20 times 1/20.

926
00:42:01,750 --> 00:42:05,260
So now in one step, we update.

927
00:42:05,260 --> 00:42:12,100
Same exact sum, we
normalize, same exact answer.

928
00:42:12,100 --> 00:42:16,300
And I mentioned that some
groups, maybe when slightly

929
00:42:16,300 --> 00:42:21,290
prompted by me, said oh,
multiplication is commutative.

930
00:42:21,290 --> 00:42:22,020
Right?

931
00:42:22,020 --> 00:42:24,390
That's all that's going on here.

932
00:42:24,390 --> 00:42:26,260
In the end, all
we're doing is taking

933
00:42:26,260 --> 00:42:29,300
our prior, the first likelihood
column, the second likelihood

934
00:42:29,300 --> 00:42:31,660
column, multiplying them
together and then normalizing

935
00:42:31,660 --> 00:42:32,180
it.

936
00:42:32,180 --> 00:42:33,182
Right?

937
00:42:33,182 --> 00:42:35,390
It doesn't matter what order
those likelihood columns

938
00:42:35,390 --> 00:42:35,680
are in.

939
00:42:35,680 --> 00:42:37,263
It doesn't matter
if we think of the 5

940
00:42:37,263 --> 00:42:39,310
and then the 9, the 9
and the 5, or the 5 and 9

941
00:42:39,310 --> 00:42:41,900
multiplied together to make
their own likelihood column.

942
00:42:41,900 --> 00:42:44,240
You get the same exact
unnormalized result no matter

943
00:42:44,240 --> 00:42:45,870
what.

944
00:42:45,870 --> 00:42:46,580
OK.

945
00:42:46,580 --> 00:42:50,380
So this is all there is
to why it doesn't matter

946
00:42:50,380 --> 00:42:53,900
what order you update
the data in sequentially,

947
00:42:53,900 --> 00:42:56,110
it just matters
what the data is.

948
00:42:56,110 --> 00:42:57,960
Now I also had a question.

949
00:42:57,960 --> 00:43:00,880
Why, if you could
do it all once,

950
00:43:00,880 --> 00:43:04,740
would you even bother with
doing it once and then

951
00:43:04,740 --> 00:43:06,280
doing it again?

952
00:43:06,280 --> 00:43:08,180
It's a very good question.

953
00:43:08,180 --> 00:43:11,079
Does anyone want to
suggest a reason?

954
00:43:11,079 --> 00:43:12,745
Maybe other than the
group that I talked

955
00:43:12,745 --> 00:43:16,594
to specifically about it Yeah.

956
00:43:16,594 --> 00:43:18,010
AUDIENCE: Well,
I'm just thinking,

957
00:43:18,010 --> 00:43:20,146
as you're saying, if you
have ongoing patients

958
00:43:20,146 --> 00:43:22,450
and you have to like
continuously update,

959
00:43:22,450 --> 00:43:25,866
then the first
data set may be all

960
00:43:25,866 --> 00:43:27,616
that you have at that
moment, but then you

961
00:43:27,616 --> 00:43:29,004
have to keeping adding on to it.

962
00:43:29,004 --> 00:43:30,420
PROFESSOR 2: That's
exactly right.

963
00:43:30,420 --> 00:43:31,370
That's exactly right.

964
00:43:31,370 --> 00:43:36,280
So, in life and in
clinical trials,

965
00:43:36,280 --> 00:43:38,680
you don't get all the
data necessarily at once.

966
00:43:38,680 --> 00:43:42,579
It sort of comes
in continuously.

967
00:43:42,579 --> 00:43:44,370
When I wake up in the
morning, I don't know

968
00:43:44,370 --> 00:43:46,900
what temperature it is outside.

969
00:43:46,900 --> 00:43:49,004
I have a prior, based
on what temperature

970
00:43:49,004 --> 00:43:50,670
it was yesterday at
about the same time,

971
00:43:50,670 --> 00:43:52,227
like I have a memory of that.

972
00:43:52,227 --> 00:43:53,810
So it gives me some
prior distribution

973
00:43:53,810 --> 00:43:55,140
on the temperature.

974
00:43:55,140 --> 00:43:57,640
OK, that might be a continuous
one, like maybe a bell curve.

975
00:43:57,640 --> 00:44:00,441
We're going to get to that next
week, these continuous updating

976
00:44:00,441 --> 00:44:00,940
problems.

977
00:44:00,940 --> 00:44:02,110
OK.

978
00:44:02,110 --> 00:44:03,590
And then, when I
get out of bed, I

979
00:44:03,590 --> 00:44:06,940
see that the window is
covered in condensation.

980
00:44:06,940 --> 00:44:08,310
That's some data.

981
00:44:08,310 --> 00:44:09,440
I can update my beliefs.

982
00:44:09,440 --> 00:44:12,790
Probably means
it's kind of cold.

983
00:44:12,790 --> 00:44:13,910
Great.

984
00:44:13,910 --> 00:44:16,640
Then I can open the
window to let the cat out,

985
00:44:16,640 --> 00:44:18,870
and a burst of freezing
cold air comes in,

986
00:44:18,870 --> 00:44:21,328
like it did this morning--
well, actually not this morning.

987
00:44:21,328 --> 00:44:22,960
This morning was great.

988
00:44:22,960 --> 00:44:25,700
Yesterday it was cold.

989
00:44:25,700 --> 00:44:28,841
And again, now I have more
data and I update my beliefs.

990
00:44:28,841 --> 00:44:29,340
All right.

991
00:44:29,340 --> 00:44:30,715
You can go through
life this way.

992
00:44:30,715 --> 00:44:32,810
It'll give you a headache,
but you can do it.

993
00:44:32,810 --> 00:44:36,070
Now, with clinical trials,
the same thing can happen.

994
00:44:36,070 --> 00:44:37,320
Maybe you do a pilot study.

995
00:44:37,320 --> 00:44:44,930
Maybe you have some belief
that eating almonds causes

996
00:44:44,930 --> 00:44:48,220
something, stomach cramps.

997
00:44:48,220 --> 00:44:48,970
Great.

998
00:44:48,970 --> 00:44:50,940
And maybe it's just for you.

999
00:44:50,940 --> 00:44:52,582
But you want to do
a study about that.

1000
00:44:52,582 --> 00:44:55,040
So maybe you know for yourself
that eating almonds give you

1001
00:44:55,040 --> 00:44:56,500
stomach cramps.

1002
00:44:56,500 --> 00:44:58,550
You're at the Harvard
School of Public Health

1003
00:44:58,550 --> 00:45:00,820
and you really want to
investigate this further,

1004
00:45:00,820 --> 00:45:03,130
so you gather a few
willing subjects,

1005
00:45:03,130 --> 00:45:05,780
you feed them some almonds, you
see if they get stomach cramps,

1006
00:45:05,780 --> 00:45:07,080
how many of them.

1007
00:45:07,080 --> 00:45:09,800
And maybe your prior
ahead was you just kind of

1008
00:45:09,800 --> 00:45:12,024
felt like you had this
really strong belief that

1009
00:45:12,024 --> 00:45:13,690
almonds cause stomach
cramps, because it

1010
00:45:13,690 --> 00:45:14,670
does for you and you
really haven't even

1011
00:45:14,670 --> 00:45:15,650
asked a lot of other people.

1012
00:45:15,650 --> 00:45:17,483
So you might have a lot
of prior probability

1013
00:45:17,483 --> 00:45:19,630
on that hypothesis being true.

1014
00:45:19,630 --> 00:45:21,060
Once you collect
a few more people

1015
00:45:21,060 --> 00:45:22,930
and maybe you ask your
friends, maybe you

1016
00:45:22,930 --> 00:45:25,054
find that most of them
don't have the same problem.

1017
00:45:25,054 --> 00:45:27,920
Suddenly your posterior
belief changes

1018
00:45:27,920 --> 00:45:30,880
and now you're not quite so
convinced that it's true.

1019
00:45:30,880 --> 00:45:32,470
And then, if you're
really ambitious

1020
00:45:32,470 --> 00:45:35,010
and you have a lot of
grant money from the NIH,

1021
00:45:35,010 --> 00:45:36,880
you recruit 1,000
people and you do

1022
00:45:36,880 --> 00:45:39,020
a randomized controlled
trial with placebo

1023
00:45:39,020 --> 00:45:43,760
and you find out that
exactly 10% of the folks

1024
00:45:43,760 --> 00:45:46,310
complain about the stomach
cramps with almonds.

1025
00:45:46,310 --> 00:45:48,069
Maybe 10% on placebo
also, so you're

1026
00:45:48,069 --> 00:45:49,360
not quite sure what's going on.

1027
00:45:49,360 --> 00:45:51,528
But the point is,
you're getting data

1028
00:45:51,528 --> 00:45:52,736
a little more, a little more.

1029
00:45:52,736 --> 00:45:57,410
And you can keep incorporating
it, updating your beliefs.

1030
00:45:57,410 --> 00:45:58,810
So that's the principle.

1031
00:45:58,810 --> 00:46:00,660
One last thing
I'll point out is,

1032
00:46:00,660 --> 00:46:02,640
what if you have a
prior belief that

1033
00:46:02,640 --> 00:46:07,210
is so strong, so you are
so sure, for example,

1034
00:46:07,210 --> 00:46:10,025
that almonds don't
cause stomach pain,

1035
00:46:10,025 --> 00:46:12,400
because it doesn't for you
and you're just sure that that

1036
00:46:12,400 --> 00:46:14,136
would be ridiculous, right?

1037
00:46:14,136 --> 00:46:15,885
So you've put your
prior probability at 0.

1038
00:46:18,557 --> 00:46:20,640
How much evidence will it
take, how much data will

1039
00:46:20,640 --> 00:46:22,680
take for you to be convinced?

1040
00:46:25,280 --> 00:46:26,150
Infinite.

1041
00:46:26,150 --> 00:46:30,480
No amount of data is going
to convince you otherwise.

1042
00:46:30,480 --> 00:46:33,580
This is why whenever you do
this kind of Bayesian Updating,

1043
00:46:33,580 --> 00:46:37,620
it's generally best to leave
a little bit of possibility

1044
00:46:37,620 --> 00:46:39,580
for Santa Claus, OK.

1045
00:46:39,580 --> 00:46:41,300
Like a little bit.

1046
00:46:41,300 --> 00:46:46,040
Just so that, should a fat
white man come down your chimney

1047
00:46:46,040 --> 00:46:48,170
and deliver the
presents one year,

1048
00:46:48,170 --> 00:46:52,300
you have a
possibility to adjust.

1049
00:46:52,300 --> 00:46:55,340
So those are some
things to keep in mind.

1050
00:46:55,340 --> 00:46:57,770
I think we've got one
more board question

1051
00:46:57,770 --> 00:46:59,860
that Jerry's going to lead.

1052
00:46:59,860 --> 00:47:05,080
PROFESSOR: Right, so it's
nice to get probabilities

1053
00:47:05,080 --> 00:47:08,110
for hypotheses, but one
thing you also want to do

1054
00:47:08,110 --> 00:47:10,850
is predict what's
going to come next.

1055
00:47:10,850 --> 00:47:15,320
And so that's what we call
probabilistic prediction.

1056
00:47:15,320 --> 00:47:19,870
If I pick one of the John's 5
die, and I'm about to roll it,

1057
00:47:19,870 --> 00:47:23,200
you could tell me using the
Law of Total Probability

1058
00:47:23,200 --> 00:47:27,740
the probability that I get a 5.

1059
00:47:27,740 --> 00:47:30,530
And we've done that, we
did that in the first unit

1060
00:47:30,530 --> 00:47:32,890
on probability in this class.

1061
00:47:32,890 --> 00:47:37,960
But suppose he rolls a 5 first.

1062
00:47:37,960 --> 00:47:41,650
Now I can update my
probabilities for which

1063
00:47:41,650 --> 00:47:43,200
die it is, which
is going to update

1064
00:47:43,200 --> 00:47:48,200
my estimation of the
probability of rolling, say a 4

1065
00:47:48,200 --> 00:47:49,779
on the next roll.

1066
00:47:49,779 --> 00:47:51,820
This is what we'll call
probabilistic prediction.

1067
00:47:51,820 --> 00:47:54,240
We can predict what's
going to happen next,

1068
00:47:54,240 --> 00:47:59,104
at least with probabilities,
based on the data

1069
00:47:59,104 --> 00:47:59,770
that we've seen.

1070
00:47:59,770 --> 00:48:03,660
We update our predictions
based on that.

1071
00:48:03,660 --> 00:48:05,220
So here's a set up.

1072
00:48:05,220 --> 00:48:06,510
D_1 is the first roll.

1073
00:48:06,510 --> 00:48:07,380
D_2 is the second.

1074
00:48:10,840 --> 00:48:12,202
I'll do this example.

1075
00:48:12,202 --> 00:48:13,410
Oh no, this a board question.

1076
00:48:13,410 --> 00:48:15,280
You're going to do this.

1077
00:48:15,280 --> 00:48:16,430
You're on your own here.

1078
00:48:16,430 --> 00:48:18,860
So first, you
don't know anything

1079
00:48:18,860 --> 00:48:23,510
about the die except that 1
of the 5 was chosen at random.

1080
00:48:23,510 --> 00:48:27,320
What's the probability that
the first roll will be a 5?

1081
00:48:27,320 --> 00:48:31,690
Now you've rolled that once,
and you're going to roll again.

1082
00:48:31,690 --> 00:48:34,030
And you can ask, well
now I've seen some data.

1083
00:48:34,030 --> 00:48:38,350
I have some information
about which die was chosen.

1084
00:48:38,350 --> 00:48:41,810
What's the probability that
I'll see a 4 on the next die.

1085
00:48:41,810 --> 00:48:45,280
So that's your job to find
that for this problem.

1086
00:48:52,368 --> 00:48:54,826
PROFESSOR 2: Cool, all right,
so this is your first answer?

1087
00:48:54,826 --> 00:48:55,492
AUDIENCE: Right.

1088
00:48:55,492 --> 00:48:58,810
And then, now we're
just multiplying across.

1089
00:48:58,810 --> 00:49:01,798
You get 0.

1090
00:49:01,798 --> 00:49:14,248
1 over 180, 1 over 320,
1 over 740, [INAUDIBLE].

1091
00:49:14,248 --> 00:49:19,726
And then add those up.

1092
00:49:19,726 --> 00:49:22,150
PROFESSOR 2: Yeah, get
out your calculator.

1093
00:49:22,150 --> 00:49:23,775
What are you dividing
by to normalize?

1094
00:49:23,775 --> 00:49:25,397
AUDIENCE: Is this--
I don't know.

1095
00:49:25,397 --> 00:49:26,480
PROFESSOR 2: That's right.

1096
00:49:26,480 --> 00:49:31,190
So this is, when you add these
up, exactly the probability

1097
00:49:31,190 --> 00:49:32,249
of the 5, right?

1098
00:49:32,249 --> 00:49:34,040
So now, if you want to
repeat this process,

1099
00:49:34,040 --> 00:49:36,920
you need to use your
normalized posterior instead

1100
00:49:36,920 --> 00:49:37,920
of the original prior.

1101
00:49:37,920 --> 00:49:38,826
AUDIENCE: That's what we were--

1102
00:49:38,826 --> 00:49:39,732
PROFESSOR 2: Yeah.

1103
00:49:39,732 --> 00:49:41,550
Great.

1104
00:49:41,550 --> 00:49:43,100
Now, you just need
to do division.

1105
00:49:43,100 --> 00:49:46,140
That I can't help you with.

1106
00:49:46,140 --> 00:49:48,780
Calculator.

1107
00:49:48,780 --> 00:49:50,226
Division skills, either way.

1108
00:49:58,207 --> 00:49:59,040
[INTERPOSING VOICES]

1109
00:50:13,597 --> 00:50:15,555
PROFESSOR 2: Now, is this
the probability of 4,

1110
00:50:15,555 --> 00:50:18,184
or the probability of 4 given 5?

1111
00:50:18,184 --> 00:50:18,725
That's right.

1112
00:50:21,350 --> 00:50:25,700
So tell me what you
did to get these.

1113
00:50:25,700 --> 00:50:28,676
AUDIENCE: So this
column [INAUDIBLE]

1114
00:50:28,676 --> 00:50:31,156
which we calculated
for, and then you

1115
00:50:31,156 --> 00:50:34,628
sum it up to get the total
probability of getting a 5.

1116
00:50:34,628 --> 00:50:36,116
And then [INAUDIBLE]
normalized it,

1117
00:50:36,116 --> 00:50:40,100
and then took the new
[INAUDIBLE] times the same--

1118
00:50:40,100 --> 00:50:41,475
PROFESSOR 2: The
same likelihood.

1119
00:50:41,475 --> 00:50:43,183
AUDIENCE: And then
you get the new column

1120
00:50:43,183 --> 00:50:44,265
and then you add it up.

1121
00:50:44,265 --> 00:50:46,425
[INAUDIBLE]

1122
00:50:46,425 --> 00:50:47,300
PROFESSOR 2: Perfect.

1123
00:50:47,300 --> 00:50:49,832
Now you said, just for the
heck of it you normalized it,

1124
00:50:49,832 --> 00:50:52,040
but it's actually really
important that you normalize

1125
00:50:52,040 --> 00:50:54,060
it, because if you're going
to use the Law of Total

1126
00:50:54,060 --> 00:50:55,640
Probability again,
right, you've got

1127
00:50:55,640 --> 00:50:58,170
to be working with a probability
mass function, something

1128
00:50:58,170 --> 00:50:59,362
that adds to 1 here.

1129
00:50:59,362 --> 00:51:01,320
If you didn't normalize,
then your final answer

1130
00:51:01,320 --> 00:51:03,300
would get multiplied
by something, right?

1131
00:51:03,300 --> 00:51:05,220
Whatever that
normalization constant was,

1132
00:51:05,220 --> 00:51:05,710
or its reciprocal.

1133
00:51:05,710 --> 00:51:06,210
Perfect.

1134
00:51:12,955 --> 00:51:13,830
Check these guys out.

1135
00:51:20,110 --> 00:51:23,617
PROFESSOR: And this right
here is not a probability,

1136
00:51:23,617 --> 00:51:24,700
because it's unnormalized.

1137
00:51:24,700 --> 00:51:26,670
So what should you
normalize this by?

1138
00:51:26,670 --> 00:51:29,230
AUDIENCE: [INAUDIBLE]

1139
00:51:29,230 --> 00:51:29,920
PROFESSOR: Yeah.

1140
00:51:29,920 --> 00:51:32,410
In other words, if you made a
normalized probability here,

1141
00:51:32,410 --> 00:51:34,810
it would be this divided by
that, this divided by that.

1142
00:51:34,810 --> 00:51:36,010
AUDIENCE: So all we need
to do is divide this--

1143
00:51:36,010 --> 00:51:37,051
PROFESSOR: Exactly right.

1144
00:51:54,042 --> 00:51:55,208
PROFESSOR 2: How's it going?

1145
00:51:55,208 --> 00:51:57,580
AUDIENCE: It's going fine.

1146
00:51:57,580 --> 00:51:58,587
So we're just saying--

1147
00:51:58,587 --> 00:51:59,420
[INTERPOSING VOICES]

1148
00:52:17,379 --> 00:52:19,170
PROFESSOR 2: So, what
did you get for this?

1149
00:52:19,170 --> 00:52:20,490
How did you find it?

1150
00:52:20,490 --> 00:52:23,480
So you summed up this, right?

1151
00:52:23,480 --> 00:52:26,189
Great, and then what
did you do next?

1152
00:52:26,189 --> 00:52:27,730
AUDIENCE: We do the
same thing for 4.

1153
00:52:27,730 --> 00:52:29,354
PROFESSOR 2: So,
starting from scratch.

1154
00:52:29,354 --> 00:52:31,940
AUDIENCE: Yeah, starting
from scratch if you roll a 4.

1155
00:52:31,940 --> 00:52:32,800
AUDIENCE: OK.

1156
00:52:32,800 --> 00:52:38,840
So how does that incorporate the
fact that you first rolled a 5?

1157
00:52:38,840 --> 00:52:41,429
AUDIENCE: So we're trying to
go back to the original Bayes,

1158
00:52:41,429 --> 00:52:43,470
but that didn't work out
so well because we don't

1159
00:52:43,470 --> 00:52:44,677
know how to calculate this.

1160
00:52:44,677 --> 00:52:46,010
PROFESSOR 2: Exactly, all right.

1161
00:52:46,010 --> 00:52:47,990
So this is no easier than that.

1162
00:52:47,990 --> 00:52:54,230
So here's the idea: so you
start with this belief, right?

1163
00:52:54,230 --> 00:52:55,660
And you have your likelihood.

1164
00:52:55,660 --> 00:52:57,493
You multiply them, Law
of Total Probability,

1165
00:52:57,493 --> 00:52:59,850
and you get the
probability of rolling a 5.

1166
00:52:59,850 --> 00:53:03,070
Now at that point, after
you've rolled this 5,

1167
00:53:03,070 --> 00:53:05,070
your beliefs have
changed, based on that,

1168
00:53:05,070 --> 00:53:08,370
about the probability that
you have each of the 5 dice.

1169
00:53:08,370 --> 00:53:12,500
So if we take that and
you use that as your prior

1170
00:53:12,500 --> 00:53:16,320
here, then the likelihood,
multiply and sum,

1171
00:53:16,320 --> 00:53:18,304
that should give you--

1172
00:53:18,304 --> 00:53:19,220
AUDIENCE: I like that.

1173
00:53:19,220 --> 00:53:20,130
PROFESSOR 2: OK?

1174
00:53:20,130 --> 00:53:21,857
AUDIENCE: Oh, but
now we have decimals.

1175
00:53:21,857 --> 00:53:23,190
AUDIENCE: Now you have decimals.

1176
00:53:23,190 --> 00:53:24,555
I can't make your life too easy.

1177
00:53:31,093 --> 00:53:31,926
AUDIENCE: Use these.

1178
00:53:35,430 --> 00:53:38,010
PROFESSOR: Right, so
you multiply this number

1179
00:53:38,010 --> 00:53:39,984
by this, this number by this.

1180
00:53:39,984 --> 00:53:41,900
So in the end, you'll
get all of these numbers

1181
00:53:41,900 --> 00:53:44,440
except they're all
multiplied by, divided by.

1182
00:53:44,440 --> 00:53:48,540
So you should divide this
by-- to get a probability.

1183
00:53:48,540 --> 00:53:50,940
That make sense?

1184
00:53:50,940 --> 00:53:55,010
So you have to be
careful for figuring out

1185
00:53:55,010 --> 00:53:57,420
probabilities of
hypotheses, we can

1186
00:53:57,420 --> 00:53:59,760
avoid using the unnormalized.

1187
00:53:59,760 --> 00:54:01,450
We don't have to
keep normalizing.

1188
00:54:01,450 --> 00:54:03,890
But when you want to do
posterior prediction,

1189
00:54:03,890 --> 00:54:07,280
you want real
probabilities of data.

1190
00:54:07,280 --> 00:54:09,280
So you're going to
have to normalize

1191
00:54:09,280 --> 00:54:11,700
these numbers at some point.

1192
00:54:11,700 --> 00:54:13,384
Either first or at the end.

1193
00:54:13,384 --> 00:54:15,300
It's probably, if you're
going to do it a lot,

1194
00:54:15,300 --> 00:54:16,345
easier to just keep doing it.

1195
00:54:16,345 --> 00:54:17,220
AUDIENCE: OK, Thanks.

1196
00:54:30,700 --> 00:54:32,182
AUDIENCE: How are we doing?

1197
00:54:32,182 --> 00:54:33,140
PROFESSOR: You tell me.

1198
00:54:36,090 --> 00:54:37,520
It's a beautiful use of color.

1199
00:54:40,339 --> 00:54:41,172
[INTERPOSING VOICES]

1200
00:54:48,735 --> 00:54:50,860
PROFESSOR: So is this
normalized or not normalized?

1201
00:54:50,860 --> 00:54:51,980
AUDIENCE: This is normalized.

1202
00:54:51,980 --> 00:54:52,896
PROFESSOR: Normalized.

1203
00:54:52,896 --> 00:54:54,960
Good, because you want
real probabilities there.

1204
00:54:54,960 --> 00:54:55,670
Yes, thank you.

1205
00:54:55,670 --> 00:54:56,545
AUDIENCE: [INAUDIBLE]

1206
00:54:59,553 --> 00:55:00,928
AUDIENCE: And so
then we multiply

1207
00:55:00,928 --> 00:55:03,400
by what are the likelihoods.

1208
00:55:03,400 --> 00:55:04,234
PROFESSOR: Yeah.

1209
00:55:04,234 --> 00:55:06,646
AUDIENCE: [INAUDIBLE]

1210
00:55:06,646 --> 00:55:08,520
PROFESSOR: And now you're
going to sum it up.

1211
00:55:08,520 --> 00:55:09,452
AUDIENCE: Yeah.

1212
00:55:09,452 --> 00:55:10,850
PROFESSOR: That's perfect.

1213
00:55:10,850 --> 00:55:12,222
AUDIENCE: Way to go Caitlin!

1214
00:55:12,222 --> 00:55:13,680
PROFESSOR: Because
the Law of Total

1215
00:55:13,680 --> 00:55:16,044
Probabilities says
multiply this by this.

1216
00:55:16,044 --> 00:55:17,710
That's just what you
did with the trees,

1217
00:55:17,710 --> 00:55:19,460
do you see the trees
when I wave my hands?

1218
00:55:24,452 --> 00:55:24,952
Beautiful.

1219
00:55:24,952 --> 00:55:25,827
AUDIENCE: [INAUDIBLE]

1220
00:55:32,182 --> 00:55:34,700
AUDIENCE: Our new prior is
given the data we had before.

1221
00:55:34,700 --> 00:55:35,325
PROFESSOR: Yes.

1222
00:55:40,210 --> 00:55:42,182
That's good.

1223
00:55:42,182 --> 00:55:44,880
PROFESSOR 2: You're
adding them up?

1224
00:55:44,880 --> 00:55:46,170
OK, so that looks right.

1225
00:55:49,050 --> 00:55:51,530
So let's think, how
does this number

1226
00:55:51,530 --> 00:55:53,900
compare to what you
would have, if you didn't

1227
00:55:53,900 --> 00:55:55,230
know this other information?

1228
00:55:55,230 --> 00:55:59,250
So suppose you knew
nothing going in,

1229
00:55:59,250 --> 00:56:03,169
do you think that the fact that
you rolled a 5 the first time

1230
00:56:03,169 --> 00:56:04,710
makes it more or
less likely that you

1231
00:56:04,710 --> 00:56:05,840
roll a 4 the second time.

1232
00:56:05,840 --> 00:56:07,329
AUDIENCE: It should
be less likely.

1233
00:56:07,329 --> 00:56:08,620
PROFESSOR 2: Is it less likely?

1234
00:56:08,620 --> 00:56:10,580
AUDIENCE: Because you
get rid of the 4--

1235
00:56:10,580 --> 00:56:13,280
the die has only 4 sides.

1236
00:56:13,280 --> 00:56:15,370
PROFESSOR 2: So it might
make you go down, right?

1237
00:56:15,370 --> 00:56:17,703
On the other hand, you're
going to give more probability

1238
00:56:17,703 --> 00:56:20,660
to the 6-sided, let's say
that, than it had originally.

1239
00:56:20,660 --> 00:56:22,160
So they're sort of
competing forces.

1240
00:56:22,160 --> 00:56:23,201
So I'm actually not sure.

1241
00:56:23,201 --> 00:56:25,239
I'm not actually sure
which is the right answer.

1242
00:56:25,239 --> 00:56:27,697
But anyway, it's something you
can easily check or compute.

1243
00:56:30,704 --> 00:56:32,120
PROFESSOR: This
is a standard day.

1244
00:56:32,120 --> 00:56:33,880
We always finish ahead of time.

1245
00:56:38,480 --> 00:56:40,055
I don't need that right here.

1246
00:56:46,020 --> 00:56:48,350
Like with all the other
answers today, we'll

1247
00:56:48,350 --> 00:56:52,130
use the screens
to show the table.

1248
00:56:54,850 --> 00:56:58,540
So when we go through it, we do
the usual thing we did before.

1249
00:56:58,540 --> 00:57:02,030
We started with a
5, so we'll update

1250
00:57:02,030 --> 00:57:05,860
our probabilities of the
hypotheses based on that 5.

1251
00:57:05,860 --> 00:57:08,970
So that's the first
column is the prior, then

1252
00:57:08,970 --> 00:57:12,260
the likelihood of
getting a 5, which

1253
00:57:12,260 --> 00:57:16,850
is 0, if you have the 4-sided
die, 1/6, 1/8, 1/12, and 1/20.

1254
00:57:16,850 --> 00:57:18,810
We get our
unnormalized posterior.

1255
00:57:18,810 --> 00:57:20,580
And then in this
table, I actually

1256
00:57:20,580 --> 00:57:22,480
got the normalized
posterior, because when

1257
00:57:22,480 --> 00:57:25,570
you're computing
predictive probabilities,

1258
00:57:25,570 --> 00:57:27,790
you need to use
normalized probabilities.

1259
00:57:27,790 --> 00:57:30,430
They should be
true probabilities.

1260
00:57:30,430 --> 00:57:35,750
So we use this column here
of the posterior probability.

1261
00:57:35,750 --> 00:57:38,160
There's different orders you
could do this calculation.

1262
00:57:38,160 --> 00:57:42,700
I think this is one of
the straightforward ones.

1263
00:57:42,700 --> 00:57:50,400
Then we look at the likelihood
of D_2 in the next column.

1264
00:57:50,400 --> 00:57:52,500
Now I put a star for
H_4 because there's

1265
00:57:52,500 --> 00:57:56,990
no way you could get
a probability there

1266
00:57:56,990 --> 00:57:59,180
if you have D_1 and D_2.

1267
00:57:59,180 --> 00:58:01,880
If I had left off the
D_1, which most people did

1268
00:58:01,880 --> 00:58:04,580
and which is OK because
of the independence

1269
00:58:04,580 --> 00:58:07,100
John talked about--
given the hypothesis

1270
00:58:07,100 --> 00:58:09,220
the rolls are independent--
I would have had

1271
00:58:09,220 --> 00:58:11,230
a 1/4 where I have the star.

1272
00:58:11,230 --> 00:58:14,990
But I'm going to multiply it
by 0 anyway, so who cares.

1273
00:58:14,990 --> 00:58:19,580
So I have my new
priors-- excuse me,

1274
00:58:19,580 --> 00:58:23,210
my new likelihoods for
the second bit of data,

1275
00:58:23,210 --> 00:58:27,060
which are the same as the old
ones: 1/6, 1/8, 1/12, 1/20.

1276
00:58:27,060 --> 00:58:30,000
Now my total probability
is done the same way.

1277
00:58:30,000 --> 00:58:33,200
You multiply in this
new language posterior

1278
00:58:33,200 --> 00:58:37,610
1, which is your prior
for likelihood 2,

1279
00:58:37,610 --> 00:58:42,810
times likelihood 2 and sum them
up, and I get this last column.

1280
00:58:47,720 --> 00:58:51,730
So again, once I have
my new, my posterior

1281
00:58:51,730 --> 00:58:53,470
for the first bit
of data, I can use

1282
00:58:53,470 --> 00:58:56,210
that in the Law of
Total Probability

1283
00:58:56,210 --> 00:58:57,751
to make a prediction.

1284
00:58:57,751 --> 00:58:59,000
So what's happened right here?

1285
00:58:59,000 --> 00:59:00,230
What's become more likely?

1286
00:59:05,120 --> 00:59:10,240
This 0.124 seems a
little bit larger

1287
00:59:10,240 --> 00:59:12,460
than I would have
guessed ahead of time.

1288
00:59:15,790 --> 00:59:18,016
What's happened with
rolling a 5 and then a 4

1289
00:59:18,016 --> 00:59:19,390
to make that a
little bit bigger?

1290
00:59:25,620 --> 00:59:26,120
Any ideas?

1291
00:59:33,700 --> 00:59:35,340
AUDIENCE: When you
roll a 5 first,

1292
00:59:35,340 --> 00:59:37,252
it increases the
probability that it's

1293
00:59:37,252 --> 00:59:38,836
one of the lower dice.

1294
00:59:38,836 --> 00:59:39,710
PROFESSOR: Excellent.

1295
00:59:39,710 --> 00:59:41,578
AUDIENCE: So then it
makes it more likely

1296
00:59:41,578 --> 00:59:42,925
that you'll get a 4.

1297
00:59:42,925 --> 00:59:43,800
PROFESSOR: Excellent.

1298
00:59:43,800 --> 00:59:45,850
That's what I would think too.

1299
00:59:45,850 --> 00:59:48,400
Seeing a 5-- a 4,
what did roll first?

1300
00:59:48,400 --> 00:59:51,820
The 5-- makes the
smaller die more likely.

1301
00:59:51,820 --> 00:59:53,540
If the smaller die
is more likely,

1302
00:59:53,540 --> 00:59:57,170
then you're more likely
to roll a 4 and a 5.

1303
00:59:57,170 --> 01:00:00,780
For instance, if I
had the 6-sided die,

1304
01:00:00,780 --> 01:00:02,960
what's the probability
I'd roll a 4 and a 5

1305
01:00:02,960 --> 01:00:04,150
with the 6-sided die?

1306
01:00:07,520 --> 01:00:11,030
1 in 36 which is about 3%.

1307
01:00:11,030 --> 01:00:13,010
So this doesn't get
up to 3%, but it's

1308
01:00:13,010 --> 01:00:15,610
a little bigger than
if I had the 20-sided

1309
01:00:15,610 --> 01:00:19,045
die where it would
be-- what's 1/400.

1310
01:00:19,045 --> 01:00:20,380
PROFESSOR 2: Very small.

1311
01:00:20,380 --> 01:00:23,030
PROFESSOR: Small,
John says it's small.

1312
01:00:23,030 --> 01:00:25,580
PROFESSOR 2: So, this is one
argument you can make, right?

1313
01:00:25,580 --> 01:00:30,470
That rolling the 5 makes it
likely that it's a smaller die,

1314
01:00:30,470 --> 01:00:32,290
and therefore more
likely to roll a 4 next.

1315
01:00:32,290 --> 01:00:34,380
But is there
another argument you

1316
01:00:34,380 --> 01:00:37,009
can make for why
maybe rolling the 5

1317
01:00:37,009 --> 01:00:38,300
would have the opposite effect?

1318
01:00:38,300 --> 01:00:42,910
It might make it less
likely to roll a 4 next?

1319
01:00:42,910 --> 01:00:45,776
AUDIENCE: My initial
thought was if you roll a 5,

1320
01:00:45,776 --> 01:00:47,950
you know it's not
the 4-sided die.

1321
01:00:47,950 --> 01:00:52,207
And that's your die that
you're most likely to roll a 4.

1322
01:00:52,207 --> 01:00:53,290
PROFESSOR 2: That's right.

1323
01:00:53,290 --> 01:00:56,880
So I'm not as sure as you as to
whether this goes up or down.

1324
01:00:56,880 --> 01:00:59,680
I think there are two
competing arguments.

1325
01:00:59,680 --> 01:01:02,190
That on the one hand,
it makes it more likely

1326
01:01:02,190 --> 01:01:05,150
it's a 6 or 8-sided, but it also
makes it completely impossible

1327
01:01:05,150 --> 01:01:06,005
that it's a 4-sided.

1328
01:01:06,005 --> 01:01:08,213
PROFESSOR: I'll give you 3
to 1 odds that it goes up.

1329
01:01:08,213 --> 01:01:09,834
PROFESSOR 2: Aw crap.

1330
01:01:09,834 --> 01:01:11,250
PROFESSOR: So what
he doesn't know

1331
01:01:11,250 --> 01:01:13,550
is whether I've computed
it already or not.

1332
01:01:13,550 --> 01:01:15,770
PROFESSOR 2: He probably has.

1333
01:01:15,770 --> 01:01:18,730
But he probably did it at like
3 AM, so I don't trust it.

1334
01:01:18,730 --> 01:01:20,530
PROFESSOR: I simulated it in R.

1335
01:01:20,530 --> 01:01:22,460
PROFESSOR 2: I see.

1336
01:01:22,460 --> 01:01:23,869
Are there any questions?

1337
01:01:23,869 --> 01:01:25,660
That's what we have
for today, so are there

1338
01:01:25,660 --> 01:01:27,340
any questions about
Bayesian Updating

1339
01:01:27,340 --> 01:01:28,506
or doing these computations?

1340
01:01:32,190 --> 01:01:34,240
All right, great.