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PROFESSOR: OK, I guess we're all
set for getting close to

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00:00:27,360 --> 00:00:31,550
the end, coming now to a race
about whether we could say

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00:00:31,550 --> 00:00:35,140
anything meaningful about
Martingales or not.

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But I think we can.

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00:00:37,370 --> 00:00:43,040
I want to spend a little time
reviewing the Wald identity

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00:00:43,040 --> 00:00:47,380
today and also sequential
tests.

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00:00:47,380 --> 00:00:51,446
It turns out that last
time on the slides--

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00:00:55,750 --> 00:00:57,690
I didn't get the thresholds
confused--

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00:00:57,690 --> 00:01:01,760
I got hypothesis 0 and
hypothesis 1 interchanged from

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00:01:01,760 --> 00:01:03,560
the way we usually do them.

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00:01:03,560 --> 00:01:05,780
And it doesn't make
any difference.

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00:01:05,780 --> 00:01:11,520
There's no difference between
hypothesis 0 and hypothesis 1.

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00:01:11,520 --> 00:01:13,900
And you can do it either
way you want to.

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00:01:13,900 --> 00:01:18,000
But it gets very confusing when
you switch from one to

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00:01:18,000 --> 00:01:20,520
the other when you're halfway
through an argument.

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00:01:20,520 --> 00:01:25,960
So I'm going to go through
part of that again today.

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00:01:25,960 --> 00:01:30,370
And we will get revised slides
on the web, so that if you

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00:01:30,370 --> 00:01:37,310
want to see them with the
hypotheses done in a

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00:01:37,310 --> 00:01:40,690
consistent way, you
will see it there.

27
00:01:40,690 --> 00:01:44,750
That should be on there by
this afternoon I hope.

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00:01:44,750 --> 00:01:52,590
OK, so let's go on and review
what Mr. Wald said.

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00:01:55,530 --> 00:01:58,820
He was talking about
a random walk.

30
00:01:58,820 --> 00:02:02,910
Random walk consists of a bunch
of a sequence of IID

31
00:02:02,910 --> 00:02:04,510
random variables.

32
00:02:04,510 --> 00:02:08,949
The random walk consists of the
sequence of partial sums

33
00:02:08,949 --> 00:02:11,100
of those random variables.

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00:02:11,100 --> 00:02:16,260
And the question is if this
random walk is taking place

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00:02:16,260 --> 00:02:21,000
and you have two thresholds, one
at alpha and one of beta--

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00:02:21,000 --> 00:02:24,080
and beta is below 0 and
alpha is above 0--

37
00:02:24,080 --> 00:02:29,510
and you start at 0, of course,
and the question is when do

38
00:02:29,510 --> 00:02:33,550
you cross one of these two
thresholds and which

39
00:02:33,550 --> 00:02:34,790
threshold do cross?

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00:02:34,790 --> 00:02:38,190
What's the probability of
crossing each other, and

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00:02:38,190 --> 00:02:41,090
everything else you can say
about this problem.

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00:02:41,090 --> 00:02:44,990
And it turns out this is a very
major problem as far as

43
00:02:44,990 --> 00:02:49,300
stochastic processes are
concerned, because it comes up

44
00:02:49,300 --> 00:02:50,670
almost everywhere.

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00:02:50,670 --> 00:02:54,210
It certainly comes up as far as

46
00:02:54,210 --> 00:02:57,110
hypothesis testing is concerned.

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00:02:57,110 --> 00:02:59,940
It's probably the major problem
there when you get

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00:02:59,940 --> 00:03:02,710
into sequential analysis.

49
00:03:02,710 --> 00:03:05,090
It's the major problem there.

50
00:03:05,090 --> 00:03:07,060
So it's a very important
problem.

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00:03:07,060 --> 00:03:14,820
And what Wald said was if you
let the random variable J be

52
00:03:14,820 --> 00:03:18,960
the stopping time of this random
walk, namely, the time

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00:03:18,960 --> 00:03:23,340
in which the walk first crosses
either alpha or

54
00:03:23,340 --> 00:03:29,090
crosses the beta, and then he
said no matter what r you

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00:03:29,090 --> 00:03:34,100
choose and the range of points
where the moment generating

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00:03:34,100 --> 00:03:38,525
function g sub X of r exists.

57
00:03:38,525 --> 00:03:43,980
Y You can pick any r in that
range, and then what you get

58
00:03:43,980 --> 00:03:48,910
is this strange looking
equality here.

59
00:03:48,910 --> 00:03:58,030
And I pointed out last time it
just wasn't all that strange,

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00:03:58,030 --> 00:04:02,720
because if instead of using the
stopping time of when you

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00:04:02,720 --> 00:04:07,130
cross a threshold, if instead
you used as a stopping time

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00:04:07,130 --> 00:04:08,640
just some particular end.

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00:04:08,640 --> 00:04:11,660
You go for some number of steps,
and then you stop.

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00:04:11,660 --> 00:04:13,440
And at that point, you
have the expected

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00:04:13,440 --> 00:04:17,560
value of E to the rsn.

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00:04:17,560 --> 00:04:22,460
The expected value of E to the
rsn by definition is the

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00:04:22,460 --> 00:04:30,150
moment generating function at r
of S sub n, which is exactly

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00:04:30,150 --> 00:04:41,460
equal to the minus J times
gamma n times gamma of r.

69
00:04:41,460 --> 00:04:46,540
So all we're doing here,
all that Wald did--

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00:04:46,540 --> 00:04:48,820
as it turns out, it
was quite a bit--

71
00:04:48,820 --> 00:04:52,430
was to say that when you replace
a fixed end with a

72
00:04:52,430 --> 00:04:56,150
stopping time, you still
get the same result.

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00:04:56,150 --> 00:04:59,750
We're stating it here just for
the case of two thresholds.

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00:04:59,750 --> 00:05:01,990
Wald stated it in much
general terms.

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00:05:01,990 --> 00:05:05,620
We'll use it a more general
terms, when we say more about

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00:05:05,620 --> 00:05:06,870
martingales.

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00:05:11,740 --> 00:05:15,520
X, now remember is the
underlying random variable.

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00:05:15,520 --> 00:05:18,038
S sub n is the sum of the X's.

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00:05:29,740 --> 00:05:39,080
If X bar is less than 0, and
if gamma r star equals 0, r

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00:05:39,080 --> 00:05:43,140
star is the r at which
gamma of r equals 0.

81
00:05:43,140 --> 00:05:46,160
It's the second root
of gamma of r.

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00:05:46,160 --> 00:06:01,660
Gamma of r, if you remember,
looks like this.

83
00:06:01,660 --> 00:06:04,230
This is r star here.

84
00:06:07,610 --> 00:06:13,420
This is the expected value
of X as the slope here.

85
00:06:13,420 --> 00:06:18,930
And we're assuming that X bar
is less than 0 for this.

86
00:06:18,930 --> 00:06:22,380
I don't know how that greater
than 0 got in.

87
00:06:22,380 --> 00:06:26,060
And then what it says is the
probability that SJ is greater

88
00:06:26,060 --> 00:06:28,810
than or equal to alpha is less
than or equal to the minus

89
00:06:28,810 --> 00:06:30,880
alpha r star.

90
00:06:30,880 --> 00:06:33,080
And last time, remember, we
went through a long, messy

91
00:06:33,080 --> 00:06:35,850
bunch of equations for that.

92
00:06:35,850 --> 00:06:40,520
I looked at it again, and this
is just the simple old Markov

93
00:06:40,520 --> 00:06:43,090
inequality again.

94
00:06:43,090 --> 00:06:48,620
All you do to get this is you
say, OK, think of the random

95
00:06:48,620 --> 00:06:51,730
variable, E to the r SJ.

96
00:06:51,730 --> 00:06:54,220
SJ is a very complicated random
variable, but it's a

97
00:06:54,220 --> 00:06:56,420
random variable, nonetheless.

98
00:06:56,420 --> 00:07:02,960
So either the rSJ is a random
variable and the expected

99
00:07:02,960 --> 00:07:18,960
value of that random variable
is at r star, the expected

100
00:07:18,960 --> 00:07:21,165
value of it is just one.

101
00:07:33,430 --> 00:07:34,790
I'll write down.

102
00:07:34,790 --> 00:07:37,610
It'll be easier.

103
00:07:37,610 --> 00:07:47,630
The expected value e to the r
star S sub J is equal to 1.

104
00:07:47,630 --> 00:08:00,360
And therefore, the probability
that E to the r star SJ is a

105
00:08:00,360 --> 00:08:10,520
greater than or equal to E to
the r star alpha is just less

106
00:08:10,520 --> 00:08:31,820
than or equal to 1 over E
to the r star alpha, OK?

107
00:08:31,820 --> 00:08:33,480
And that's what the
inequality says.

108
00:08:33,480 --> 00:08:36,960
So that's all there is to it.

109
00:08:36,960 --> 00:08:38,700
OK, what?

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00:08:38,700 --> 00:08:43,155
AUDIENCE: I don't really see
why these two [INAUDIBLE]?

111
00:08:43,155 --> 00:08:47,120
They don't [INAUDIBLE].

112
00:08:47,120 --> 00:08:49,610
PROFESSOR: You need x1 negative
so that you get

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00:08:49,610 --> 00:08:53,760
another root so that
r star exists.

114
00:08:53,760 --> 00:08:58,410
If r star is positive, if the
expected value of x is

115
00:08:58,410 --> 00:09:03,090
positive, then r star is down
here at negative r.

116
00:09:13,000 --> 00:09:14,170
I mean, you're talking
about the other

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00:09:14,170 --> 00:09:15,790
threshold in a sense.

118
00:09:19,900 --> 00:09:26,050
OK, this is valid for all
lower thresholds.

119
00:09:26,050 --> 00:09:28,850
And it's also valid
for no threshold.

120
00:09:28,850 --> 00:09:32,760
OK, in other words, this
equation here does not have

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00:09:32,760 --> 00:09:34,630
beta in it at all.

122
00:09:34,630 --> 00:09:41,130
So this equation is an upper
bound on the probability, that

123
00:09:41,130 --> 00:09:43,960
you're going to cross that
threshold of alpha.

124
00:09:43,960 --> 00:09:47,550
And that upper bound is valid,
no matter where you put the

125
00:09:47,550 --> 00:09:49,480
lower bound at all.

126
00:09:49,480 --> 00:09:51,830
So you can go to the
limit as the lower

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00:09:51,830 --> 00:09:53,930
bound goes to infinity.

128
00:09:53,930 --> 00:09:57,120
And this inequality should
still be valid.

129
00:09:57,120 --> 00:10:00,200
You have a homework problem
where you actually prove that.

130
00:10:00,200 --> 00:10:04,050
Sometimes when things go to
infinity, funny things happen.

131
00:10:04,050 --> 00:10:07,790
And that proves that nothing
funny happens then.

132
00:10:07,790 --> 00:10:13,900
So what happens then is the
probability that you ever

133
00:10:13,900 --> 00:10:18,000
cross a threshold at plus alpha,
when you have a random

134
00:10:18,000 --> 00:10:26,840
variable, which has a negative
mean, is this exponent here.

135
00:10:26,840 --> 00:10:31,900
And we also sort of showed by
looking at the turn off bound

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00:10:31,900 --> 00:10:34,300
that this bound is
pretty tight.

137
00:10:34,300 --> 00:10:37,130
So in other words, what this
is saying is when you're

138
00:10:37,130 --> 00:10:40,740
looking at threshold
crossing problems--

139
00:10:40,740 --> 00:10:45,560
this quantity here, this
quantity where the second root

140
00:10:45,560 --> 00:10:48,220
of gamma of r is--

141
00:10:48,220 --> 00:10:50,460
that's sort of the
crucial parameter

142
00:10:50,460 --> 00:10:51,470
that you want to know.

143
00:10:51,470 --> 00:10:54,170
Usually the first thing you want
to know about a random

144
00:10:54,170 --> 00:10:57,150
variable is its mean, its
variance, all sorts

145
00:10:57,150 --> 00:10:58,640
of things like that.

146
00:10:58,640 --> 00:11:01,160
This is saying if you're
interested in thresholds,

147
00:11:01,160 --> 00:11:05,702
forget about all those things,
look at r star.

148
00:11:05,702 --> 00:11:09,320
If r star is positive that means
it means is negative, so

149
00:11:09,320 --> 00:11:10,580
there's no problem there.

150
00:11:10,580 --> 00:11:14,700
But this one quantity here is
sort of the most important

151
00:11:14,700 --> 00:11:18,140
parameter of all of
these problems.

152
00:11:18,140 --> 00:11:25,900
OK, so let's go back to look at
a hypothesis testing again,

153
00:11:25,900 --> 00:11:32,750
where we're looking at the
likelihood ratio of being the

154
00:11:32,750 --> 00:11:37,460
ratio of the density for
hypothesis 0 divided by

155
00:11:37,460 --> 00:11:39,960
hypothesis 1.

156
00:11:39,960 --> 00:11:45,480
What you get then is you observe
this sequence Y sub n.

157
00:11:45,480 --> 00:11:48,360
These are the observations
that you're taking.

158
00:11:48,360 --> 00:11:51,330
In other words, nature at the
beginning of this whole

159
00:11:51,330 --> 00:11:58,870
experiment chooses either H
equals 0 or H equals 1.

160
00:11:58,870 --> 00:12:02,370
At that point, you start
to make measurements.

161
00:12:02,370 --> 00:12:06,180
Now whether nature chooses H
equals 0 before or after or

162
00:12:06,180 --> 00:12:08,220
when doesn't make
any difference.

163
00:12:08,220 --> 00:12:12,090
The point is the experiment
consists of nature choosing

164
00:12:12,090 --> 00:12:13,980
one of these two hypotheses.

165
00:12:13,980 --> 00:12:18,590
You know all the probabilities
that exist in the world in

166
00:12:18,590 --> 00:12:19,710
this model.

167
00:12:19,710 --> 00:12:21,770
You go making these
measurements.

168
00:12:21,770 --> 00:12:23,640
All you of observe is
these measurements.

169
00:12:23,640 --> 00:12:28,830
You don't observe what the
hypothesis is, so you define

170
00:12:28,830 --> 00:12:33,760
this likelihood ratio of the
ratio of the densities of the

171
00:12:33,760 --> 00:12:42,940
vector Y for H equals 0 and the
vector Y with H equals 1.

172
00:12:42,940 --> 00:12:47,660
These quantities exist no
matter what the a priori

173
00:12:47,660 --> 00:12:49,360
probabilities of the thresholds

174
00:12:49,360 --> 00:12:51,470
are or anything else.

175
00:12:51,470 --> 00:12:55,430
Even without all of that, so
long as you have a model which

176
00:12:55,430 --> 00:13:01,680
tells you what the densities
of these observations are,

177
00:13:01,680 --> 00:13:04,110
conditional on each hypothesis,

178
00:13:04,110 --> 00:13:05,140
you can define this.

179
00:13:05,140 --> 00:13:09,070
This doesn't depend on a priori
probabilities at all.

180
00:13:09,070 --> 00:13:14,010
OK, so now you look at the
probability that H is equal to

181
00:13:14,010 --> 00:13:18,150
0, given all these observations
divided by the

182
00:13:18,150 --> 00:13:20,470
probability it's equal to 1.

183
00:13:20,470 --> 00:13:23,000
What you get here now,
you have the a priori

184
00:13:23,000 --> 00:13:26,090
probabilities, p0 over p1.

185
00:13:26,090 --> 00:13:28,880
Here is the likelihood
ratio here.

186
00:13:28,880 --> 00:13:34,610
So what you have this p0 over p1
times the likelihood ratio

187
00:13:34,610 --> 00:13:36,400
of this vector of however many

188
00:13:36,400 --> 00:13:39,585
observations you have observed.

189
00:13:39,585 --> 00:13:43,350
It's just a nice way of breaking
up the problem into

190
00:13:43,350 --> 00:13:50,780
the likelihood ratio and the
a priori probabilities.

191
00:13:50,780 --> 00:13:55,170
Incidentally, we haven't talked
about this at all, but

192
00:13:55,170 --> 00:13:59,730
there's an important idea and
all of this hypothesis testing

193
00:13:59,730 --> 00:14:03,600
of a sufficient statistic, and
what do you think a sufficient

194
00:14:03,600 --> 00:14:05,380
statistic is.

195
00:14:05,380 --> 00:14:07,630
It's anything from which
you can calculate

196
00:14:07,630 --> 00:14:10,020
the likelihood ratio.

197
00:14:10,020 --> 00:14:12,480
In other words, what we're
saying here, the point we're

198
00:14:12,480 --> 00:14:17,510
making, is that any intelligent
choice of

199
00:14:17,510 --> 00:14:23,090
hypothesis is it based on
a threshold test on the

200
00:14:23,090 --> 00:14:24,970
likelihood ratio.

201
00:14:24,970 --> 00:14:29,380
And therefore, the only thing
you can really be interested

202
00:14:29,380 --> 00:14:33,430
in in all your observations
is just what is

203
00:14:33,430 --> 00:14:35,190
the likelihood ratio?

204
00:14:35,190 --> 00:14:38,680
If they make all these 1,000
observations complicated sort

205
00:14:38,680 --> 00:14:41,680
of thing, you calculate
one number.

206
00:14:41,680 --> 00:14:44,220
And that's the only thing
you're interested in.

207
00:14:44,220 --> 00:14:47,900
And anything from which that
number could be calculated is

208
00:14:47,900 --> 00:14:49,900
a sufficient statistic.

209
00:14:49,900 --> 00:14:52,720
And anything from which it can't
be calculated you've

210
00:14:52,720 --> 00:14:56,260
thrown away some of the
information that you have.

211
00:14:56,260 --> 00:15:00,140
If you study communication and
you study detection, you study

212
00:15:00,140 --> 00:15:04,900
how to receive data that's being
sent, what you find is

213
00:15:04,900 --> 00:15:10,000
that right at the beginning,
even before you do any

214
00:15:10,000 --> 00:15:13,530
detection, even before you do
any filtering, there's some

215
00:15:13,530 --> 00:15:16,490
idea of a sufficient
statistic there.

216
00:15:16,490 --> 00:15:20,900
That's what you need in order to
calculate everything else.

217
00:15:20,900 --> 00:15:23,100
And you want to make sure
that you have that.

218
00:15:23,100 --> 00:15:26,240
So that's an important
idea there.

219
00:15:26,240 --> 00:15:34,340
OK, but anyway, the MAP rule,
which comes right from this,

220
00:15:34,340 --> 00:15:38,100
says if you have these a priori
probabilities, and

221
00:15:38,100 --> 00:15:41,380
you're trying to maximize the
probability of choosing

222
00:15:41,380 --> 00:15:44,750
correctly, what do you do?

223
00:15:44,750 --> 00:15:52,290
Well, your probability of H
equals 0 was the correct

224
00:15:52,290 --> 00:15:55,680
hypothesis, given all
the observations you

225
00:15:55,680 --> 00:15:57,440
made, is in fact this.

226
00:15:57,440 --> 00:15:59,820
The probability that H equals
1 is the correct

227
00:15:59,820 --> 00:16:01,670
hypothesis is this.

228
00:16:01,670 --> 00:16:04,300
What do you do if you want to
maximize the probability of

229
00:16:04,300 --> 00:16:05,270
being correct?

230
00:16:05,270 --> 00:16:07,340
You choose the one
which is biggest.

231
00:16:07,340 --> 00:16:11,260
In other words, what you do is
you look at this number.

232
00:16:11,260 --> 00:16:14,890
And if this number is bigger
than 1, you choose 0.

233
00:16:14,890 --> 00:16:20,290
If it's less than 1, you
choose hypothesis 1.

234
00:16:20,290 --> 00:16:23,960
And what it turns out to
is threshold of rule.

235
00:16:23,960 --> 00:16:26,280
You take this likelihood
ratio.

236
00:16:26,280 --> 00:16:30,020
You compare it with
p1 over p0.

237
00:16:30,020 --> 00:16:34,250
And in this case, you
select h equals 0.

238
00:16:34,250 --> 00:16:37,540
In this case you select
H equals 1.

239
00:16:37,540 --> 00:16:42,040
And the last time I just a 1 and
0 reversed, which is fine,

240
00:16:42,040 --> 00:16:44,620
but if you reverse them one
place, you want a reverse them

241
00:16:44,620 --> 00:16:47,550
every place.

242
00:16:47,550 --> 00:16:52,260
And every other threshold test
does something like this,

243
00:16:52,260 --> 00:16:56,920
except you replace p1 over p0
with some arbitrary threshold.

244
00:16:56,920 --> 00:16:59,590
You say whatever reason you
want to find for that

245
00:16:59,590 --> 00:17:03,880
threshold, that's the only
intelligent kind of

246
00:17:03,880 --> 00:17:06,460
test you can make.

247
00:17:06,460 --> 00:17:09,300
OK, then we define the
log-likelihood ratio of the

248
00:17:09,300 --> 00:17:12,710
logarithm of the likelihood
ratio.

249
00:17:12,710 --> 00:17:17,970
And that was nice because it
was a sum of this quantity

250
00:17:17,970 --> 00:17:20,339
related to the individual
observations.

251
00:17:20,339 --> 00:17:25,400
For each observation you really
want to know what f of

252
00:17:25,400 --> 00:17:30,870
Y given H of Y given 0,
divided by Y given 1.

253
00:17:30,870 --> 00:17:32,000
You want to divide those two.

254
00:17:32,000 --> 00:17:34,060
You want to take the
logarithm of it.

255
00:17:34,060 --> 00:17:37,130
And then you have those numbers,
and the sufficient

256
00:17:37,130 --> 00:17:41,070
statistic that you're interested
in is just a sum of

257
00:17:41,070 --> 00:17:42,530
those numbers.

258
00:17:42,530 --> 00:17:47,385
So you're looking at a sum
of IID, random variable.

259
00:17:47,385 --> 00:17:50,540
IID, why IID?

260
00:17:50,540 --> 00:17:57,650
Well, under the hypothesis that
H is equal to 1, those

261
00:17:57,650 --> 00:18:00,410
Yi's are IID.

262
00:18:00,410 --> 00:18:03,000
And therefore under the
hypothesis that

263
00:18:03,000 --> 00:18:04,250
H is equal to 1.

264
00:18:09,140 --> 00:18:12,410
Well, little Zi is just
a sample value.

265
00:18:12,410 --> 00:18:15,080
If you look at the random
variable which has these

266
00:18:15,080 --> 00:18:21,560
sampled values, Z sub i, under
the probability measure,

267
00:18:21,560 --> 00:18:27,850
corresponding to H equals 1,
those Z sub i's are IID.

268
00:18:27,850 --> 00:18:34,350
So what that says is when you
look at these sums of random

269
00:18:34,350 --> 00:18:39,780
variables, the sum of Zi from
1 to n, under hypothesis H

270
00:18:39,780 --> 00:18:43,750
equals 1, what do you get?

271
00:18:43,750 --> 00:18:46,540
You get a random walk.

272
00:18:46,540 --> 00:18:49,935
You get a sum of IID
random variables.

273
00:18:53,780 --> 00:18:58,770
If you take more observations,
S sub n just changes.

274
00:18:58,770 --> 00:19:02,020
With n changing, then you
have a larger number of

275
00:19:02,020 --> 00:19:02,840
observations.

276
00:19:02,840 --> 00:19:06,270
So the random walk goes a little
further out, and you

277
00:19:06,270 --> 00:19:08,825
might get closer to a threshold
or whatever.

278
00:19:11,990 --> 00:19:14,110
And that's what we're
trying to do here.

279
00:19:14,110 --> 00:19:20,680
OK, so the Z sub i's under the
hypothesis H equals 1, or IID,

280
00:19:20,680 --> 00:19:24,820
and the moment generating
function of the Z sub i's

281
00:19:24,820 --> 00:19:27,305
given H equals 1, is this.

282
00:19:30,460 --> 00:19:33,220
Let's be careful about this.

283
00:19:33,220 --> 00:19:41,240
The sampled values of the Z
sub i do not depend on the

284
00:19:41,240 --> 00:19:43,410
hypotheses at all.

285
00:19:43,410 --> 00:19:45,900
Namely, you make
an observation.

286
00:19:45,900 --> 00:19:49,280
You make an observation
of Y sub i.

287
00:19:49,280 --> 00:19:52,730
You calculate Z sub
i from Y sub i.

288
00:19:52,730 --> 00:19:55,910
That has nothing to do
with whether H equals

289
00:19:55,910 --> 00:19:58,270
0 or H equals 1.

290
00:19:58,270 --> 00:20:00,940
You try to calculate
this moment

291
00:20:00,940 --> 00:20:03,250
generating function, however.

292
00:20:03,250 --> 00:20:05,980
And you want to know what
the probability

293
00:20:05,980 --> 00:20:09,870
density of the Y's are.

294
00:20:09,870 --> 00:20:13,140
And you get a different
probability density for H

295
00:20:13,140 --> 00:20:16,500
equals 1, then you get on
the other hypothesis.

296
00:20:16,500 --> 00:20:19,850
If the observations behaved
the same way under both

297
00:20:19,850 --> 00:20:24,500
hypotheses, it wouldn't make
much sense to do the

298
00:20:24,500 --> 00:20:25,450
observation.

299
00:20:25,450 --> 00:20:27,580
Unless you have a government
grant, and you're trying to

300
00:20:27,580 --> 00:20:30,130
get money out of the government
instead of trying

301
00:20:30,130 --> 00:20:32,470
to do anything worthwhile.

302
00:20:32,470 --> 00:20:35,690
Under those circumstances, you
keep on making observations.

303
00:20:35,690 --> 00:20:38,560
You now perfectly well
that nothing is going

304
00:20:38,560 --> 00:20:39,830
to come from them.

305
00:20:39,830 --> 00:20:42,310
But otherwise, it's
a little silly.

306
00:20:42,310 --> 00:20:46,690
So this moment generating
function under the hypothesis

307
00:20:46,690 --> 00:20:52,150
H equals 1 is given by
this quantity here.

308
00:20:52,150 --> 00:20:57,630
And this density here is the
same as this density here.

309
00:20:57,630 --> 00:21:03,590
So you get this density to the
1 minus r power, and you get

310
00:21:03,590 --> 00:21:06,130
this density to the r power.

311
00:21:06,130 --> 00:21:09,210
So you get the product of
these two densities.

312
00:21:09,210 --> 00:21:16,200
You integrate it over Y, and
that's what gamma 1 of r is.

313
00:21:16,200 --> 00:21:18,780
Now I said that the really
important thing in all of

314
00:21:18,780 --> 00:21:23,990
these threshold problems
is what is our star?

315
00:21:23,990 --> 00:21:27,180
And for this problem,
r star is trivial.

316
00:21:27,180 --> 00:21:28,640
It's always the same.

317
00:21:28,640 --> 00:21:31,570
r star is always equal to 1.

318
00:21:31,570 --> 00:21:37,160
And the reason is when you set
r equal to 1 here, this

319
00:21:37,160 --> 00:21:39,080
quantity becomes 1.

320
00:21:39,080 --> 00:21:41,860
This quantity becomes
the density of Y

321
00:21:41,860 --> 00:21:44,370
conditional on H equals 0.

322
00:21:44,370 --> 00:21:46,990
When you integrate
that, you get 1.

323
00:21:46,990 --> 00:21:53,100
So for all of these hypothesis
testing problems, r star is

324
00:21:53,100 --> 00:21:56,150
equal to 1.

325
00:21:56,150 --> 00:22:00,050
Gamma 1 of 1 is equal to 0.

326
00:22:00,050 --> 00:22:02,405
And this is what this
curve says.

327
00:22:10,660 --> 00:22:17,220
OK, this is gamma 1 of r here.

328
00:22:17,220 --> 00:22:20,450
This curve starts out here,
negative slope.

329
00:22:20,450 --> 00:22:22,030
It comes up here.

330
00:22:22,030 --> 00:22:26,630
r star is equal to
1 in this case.

331
00:22:26,630 --> 00:22:29,565
And that's sort of the end
of the story for that.

332
00:22:33,080 --> 00:22:38,660
Now if you are doing a test with
a fixed value of n, you

333
00:22:38,660 --> 00:22:41,000
say I'm going to make
n observations, it's

334
00:22:41,000 --> 00:22:42,770
all I have time for.

335
00:22:42,770 --> 00:22:44,180
The week is over.

336
00:22:44,180 --> 00:22:46,680
I'm going on vacation
next week.

337
00:22:46,680 --> 00:22:48,580
I've got to stop this test.

338
00:22:48,580 --> 00:22:50,730
I've got to write my paper.

339
00:22:50,730 --> 00:22:52,290
Take the end test.

340
00:22:52,290 --> 00:22:54,170
You write your paper.

341
00:22:54,170 --> 00:22:56,820
And what do you do?

342
00:22:56,820 --> 00:23:01,420
You go through the optimal
tests the best you can.

343
00:23:01,420 --> 00:23:05,910
And what you find is given H
equals 1, an error is going to

344
00:23:05,910 --> 00:23:11,020
occur if the sum of random
variables, namely the

345
00:23:11,020 --> 00:23:14,800
log-likelihood likelihood ratio,
exceeds the logarithm

346
00:23:14,800 --> 00:23:16,490
of your threshold.

347
00:23:16,490 --> 00:23:20,650
OK, this is whatever threshold
you decide to establish.

348
00:23:20,650 --> 00:23:27,580
And we showed before that the
probability that S sub n is

349
00:23:27,580 --> 00:23:33,290
greater than or equal to log of
the threshold is evaluated

350
00:23:33,290 --> 00:23:37,220
as E to the n times this
quantity right here.

351
00:23:37,220 --> 00:23:40,210
The probability of error
given H equals 1 is

352
00:23:40,210 --> 00:23:41,430
this quantity here.

353
00:23:41,430 --> 00:23:45,380
Probability of error given H
equals 1 is the probability

354
00:23:45,380 --> 00:23:48,880
that the data looks like
H equals 0 was a right

355
00:23:48,880 --> 00:23:49,720
hypothesis.

356
00:23:49,720 --> 00:23:52,690
In other words, that you crossed
the threshold at plus

357
00:23:52,690 --> 00:23:55,790
alpha, instead of crossing
the threshold at beta.

358
00:23:59,710 --> 00:24:02,660
Excuse me.

359
00:24:02,660 --> 00:24:04,590
We have too many cases
here we're looking

360
00:24:04,590 --> 00:24:07,300
at, so it gets confusing.

361
00:24:07,300 --> 00:24:11,810
What I'm looking at here is
the probability that the

362
00:24:11,810 --> 00:24:16,090
log-likelihood ratio exceeds
this threshold data, whatever

363
00:24:16,090 --> 00:24:18,380
we set beta to be.

364
00:24:18,380 --> 00:24:22,230
Eta is set, depending on the
cost of making errors of both

365
00:24:22,230 --> 00:24:26,955
types on our a priori beta,
if we have any and

366
00:24:26,955 --> 00:24:28,420
all of those things.

367
00:24:28,420 --> 00:24:31,920
And the probability of error
given H equals 1 is this

368
00:24:31,920 --> 00:24:36,490
quantity here, which has the
threshold in it over there.

369
00:24:36,490 --> 00:24:41,070
We've looked at that a number
of times in a lecture.

370
00:24:41,070 --> 00:24:42,950
We looked at it in
chapter one.

371
00:24:42,950 --> 00:24:46,510
And then those, we looked
at it in chapter seven.

372
00:24:46,510 --> 00:24:50,220
And you calculate it by taking
this moment generating

373
00:24:50,220 --> 00:24:58,480
function, drawing attention to
it at the point where slope

374
00:24:58,480 --> 00:25:02,430
natural log of eta
divided by n.

375
00:25:02,430 --> 00:25:05,200
And then you take where
it comes in to this

376
00:25:05,200 --> 00:25:07,140
vertical axis here.

377
00:25:07,140 --> 00:25:13,690
And that's the exponent of the
error of probability when

378
00:25:13,690 --> 00:25:16,206
hypothesis 1 is correct.

379
00:25:16,206 --> 00:25:23,050
Now, if the hypothesis is H
equals 0 instead, at that

380
00:25:23,050 --> 00:25:29,630
point with H equals 0, the
expected value of this

381
00:25:29,630 --> 00:25:32,860
log-likelihood ratio is
going to be positive.

382
00:25:32,860 --> 00:25:37,800
The situation is going to be a
curve that comes over here,

383
00:25:37,800 --> 00:25:40,080
comes back at some point here.

384
00:25:40,080 --> 00:25:44,040
And what we've showed is that
this curve is just a

385
00:25:44,040 --> 00:25:47,630
translation of this
curve by 1.

386
00:25:47,630 --> 00:25:53,770
OK, namely if you calculate the
moment generating function

387
00:25:53,770 --> 00:25:59,450
for H equals 0, you get the same
thing that we got before.

388
00:26:01,970 --> 00:26:04,390
I'm not going to go through
all the details of this.

389
00:26:06,890 --> 00:26:09,695
Now you have 0 here
instead of 1.

390
00:26:13,740 --> 00:26:19,540
Over here you're going
to have just minus r.

391
00:26:19,540 --> 00:26:25,440
And over here you're going
to have 1 plus r.

392
00:26:25,440 --> 00:26:29,170
So this whole thing is
translated by 1.

393
00:26:29,170 --> 00:26:30,860
The action happens here.

394
00:26:30,860 --> 00:26:35,010
But if you translate it by 1
over in this direction, what

395
00:26:35,010 --> 00:26:38,590
happens is the error
of probability is

396
00:26:38,590 --> 00:26:41,160
determined by this.

397
00:26:44,220 --> 00:26:45,560
It's this.

398
00:26:45,560 --> 00:26:49,800
The exponent is this point right
here, gamma 1 of r0 plus

399
00:26:49,800 --> 00:26:53,430
1 minus r0, log of eta over n.

400
00:26:53,430 --> 00:26:57,280
And the r0 was again determined
by this point at

401
00:26:57,280 --> 00:27:00,718
which the slope is equal
to log eta over n.

402
00:27:03,240 --> 00:27:04,490
We did that before.

403
00:27:10,450 --> 00:27:13,440
Then I want to make clear you
understood it, because to

404
00:27:13,440 --> 00:27:16,100
really understand it, you have
to go through the arithmetic

405
00:27:16,100 --> 00:27:18,190
yourselves at least once.

406
00:27:18,190 --> 00:27:21,860
And you can do that easily by
following the notes, because

407
00:27:21,860 --> 00:27:25,650
it does it in almost
excruciating detail.

408
00:27:25,650 --> 00:27:29,820
So that's the argument
you get.

409
00:27:29,820 --> 00:27:36,380
We had this idea before, of the
Neyman-Pearson principle,

410
00:27:36,380 --> 00:27:40,750
which says you don't assume
a priori probabilities.

411
00:27:40,750 --> 00:27:45,450
You look at the probability of
making an error as being a

412
00:27:45,450 --> 00:27:50,160
trade off between the error you
make when H is equal to 1

413
00:27:50,160 --> 00:27:53,380
and the error you make
when H is equal to 0.

414
00:27:53,380 --> 00:27:55,270
In terms of the Chernoff
bound, this

415
00:27:55,270 --> 00:27:57,320
trade off is very clear.

416
00:28:00,700 --> 00:28:05,200
As you change the exponent that
you want to get under H

417
00:28:05,200 --> 00:28:09,810
equals 1, this point moves.

418
00:28:09,810 --> 00:28:12,330
The tangent then moves.

419
00:28:12,330 --> 00:28:14,300
And the exponent over
here moves.

420
00:28:14,300 --> 00:28:16,950
So you have this inverted
seesaw.

421
00:28:16,950 --> 00:28:20,170
And the exponent for one kind
of error is over here.

422
00:28:20,170 --> 00:28:24,790
And the exponent for the other
kind of error is over there.

423
00:28:24,790 --> 00:28:29,550
Then the next thing we said
was this is really stupid,

424
00:28:29,550 --> 00:28:32,410
unless you're going on
vacation this Friday.

425
00:28:32,410 --> 00:28:35,130
If you're not going on vacation
this Friday, if

426
00:28:35,130 --> 00:28:38,620
you're really serious about
making the right decision,

427
00:28:38,620 --> 00:28:44,210
then what you're going to do is
keep on making observations

428
00:28:44,210 --> 00:28:45,885
until you're pretty
sure you're right.

429
00:28:49,420 --> 00:28:53,620
Now somebody at the end of the
lecture last time pointed out

430
00:28:53,620 --> 00:28:59,840
something, which says that when
you do experiments and

431
00:28:59,840 --> 00:29:02,420
you keep on making observations
until you get the

432
00:29:02,420 --> 00:29:05,760
data that you want, there's
something very

433
00:29:05,760 --> 00:29:08,960
unethical about that.

434
00:29:08,960 --> 00:29:12,580
Is this that kind of
unethical behavior?

435
00:29:12,580 --> 00:29:13,830
Or is this really valid?

436
00:29:17,770 --> 00:29:21,840
Well, I claim this is valid,
because what we're doing when

437
00:29:21,840 --> 00:29:27,050
we're doing sequential testing
is we're deciding what we're

438
00:29:27,050 --> 00:29:29,390
going to do ahead of time.

439
00:29:29,390 --> 00:29:31,900
Namely, we've decided what we're
going to do is we're

440
00:29:31,900 --> 00:29:36,550
going to continue testing until
we cross a threshold and

441
00:29:36,550 --> 00:29:42,290
threshold gives us a suitable
probability of error.

442
00:29:42,290 --> 00:29:44,770
So we're not cooking
the books at all.

443
00:29:44,770 --> 00:29:47,730
What we're doing is we're
following this preset

444
00:29:47,730 --> 00:29:49,250
procedure we've set up.

445
00:29:49,250 --> 00:29:54,230
And the only question is can
we get a very small error

446
00:29:54,230 --> 00:29:59,240
probability by using a smaller
number of observations on the

447
00:29:59,240 --> 00:30:01,580
average than what we
need otherwise?

448
00:30:04,330 --> 00:30:06,415
Put it in terms of a
communication system.

449
00:30:09,060 --> 00:30:12,490
One kind of communication
system, you have to send some

450
00:30:12,490 --> 00:30:15,320
data from one point
to another.

451
00:30:15,320 --> 00:30:18,380
You're not going to get
any feedback on it.

452
00:30:18,380 --> 00:30:21,630
You've got to get the data
through the first time.

453
00:30:21,630 --> 00:30:23,140
It's got to be right.

454
00:30:23,140 --> 00:30:25,250
What are you going to do?

455
00:30:25,250 --> 00:30:27,920
You're going to send this data
a very large number of times

456
00:30:27,920 --> 00:30:31,960
or use a very powerful coding
technique on it.

457
00:30:31,960 --> 00:30:35,790
And by time it gets through,
you're going to be very sure

458
00:30:35,790 --> 00:30:38,860
you're right.

459
00:30:38,860 --> 00:30:42,780
Now a much better procedure, and
the thing which is used in

460
00:30:42,780 --> 00:30:46,600
almost all communication
systems, and the thing which

461
00:30:46,600 --> 00:30:50,220
we use as human beings all the
time, and the thing which

462
00:30:50,220 --> 00:30:53,120
control people use all the time,
the thing which almost

463
00:30:53,120 --> 00:30:57,150
everybody uses, because most of
us have common sense if we

464
00:30:57,150 --> 00:31:01,840
spend some time trying to do
these things, is instead of

465
00:31:01,840 --> 00:31:06,720
trying to get it right the first
time, we try little bit

466
00:31:06,720 --> 00:31:08,500
to get it right the
first time.

467
00:31:08,500 --> 00:31:10,770
And we make sure that if we
don't get it right the first

468
00:31:10,770 --> 00:31:13,440
time, we have some way of
finding out about it and

469
00:31:13,440 --> 00:31:16,550
getting it right the
second time.

470
00:31:21,390 --> 00:31:26,500
And in the scientific way of
looking at it, what we do is

471
00:31:26,500 --> 00:31:30,250
we decide ahead of time exactly
what our procedure is

472
00:31:30,250 --> 00:31:33,043
going to be for making
repetitions--

473
00:31:36,090 --> 00:31:39,100
something called ARQ in
communication systems, which

474
00:31:39,100 --> 00:31:41,910
means automatic repeat
request.

475
00:31:41,910 --> 00:31:47,470
It's automatic, which means
you don't try to make your

476
00:31:47,470 --> 00:31:51,660
decision depending on whether
you'd like to receive this 0

477
00:31:51,660 --> 00:31:54,400
or like to receive a 1.

478
00:31:54,400 --> 00:31:57,730
You make the decision ahead of
time that if you have a clean

479
00:31:57,730 --> 00:32:01,110
enough answer, you're
going to accept it.

480
00:32:01,110 --> 00:32:04,310
If it looks doubtful, you're
going to send it over again.

481
00:32:04,310 --> 00:32:08,740
That's exactly the same sort
of thing we're doing here.

482
00:32:08,740 --> 00:32:17,810
OK, when we do that, given H
equals 1, we again have this S

483
00:32:17,810 --> 00:32:22,400
sub n as a function of
n as a random walk.

484
00:32:22,400 --> 00:32:26,040
It's a sum of IID random
variables and

485
00:32:26,040 --> 00:32:27,640
conditional on H equals 1.

486
00:32:27,640 --> 00:32:28,950
You have a random walk.

487
00:32:28,950 --> 00:32:33,810
Conditional on H equals 1, you
have a negative slope on this

488
00:32:33,810 --> 00:32:34,530
random walk.

489
00:32:34,530 --> 00:32:38,480
The random walk starts out and
on the average is going to go

490
00:32:38,480 --> 00:32:41,380
down, and it's going to continue
going down forever.

491
00:32:41,380 --> 00:32:43,580
And if you're looking for
across some positive

492
00:32:43,580 --> 00:32:46,640
threshold, if it doesn't cross
it pretty soon, it's not going

493
00:32:46,640 --> 00:32:47,490
to cross it.

494
00:32:47,490 --> 00:32:50,230
But anyway, we have a test
which said we have some

495
00:32:50,230 --> 00:32:51,320
positive threshold.

496
00:32:51,320 --> 00:32:53,420
We have some negative
threshold.

497
00:32:53,420 --> 00:32:56,950
If we ever cross the positive
threshold, we say

498
00:32:56,950 --> 00:32:58,610
H is equal to 0.

499
00:32:58,610 --> 00:33:01,410
If we ever cross the negative
threshold, we say

500
00:33:01,410 --> 00:33:02,630
H is equal to 1.

501
00:33:02,630 --> 00:33:05,600
And then we're done with it.

502
00:33:05,600 --> 00:33:10,200
OK, now, let me give you another
argument why that

503
00:33:10,200 --> 00:33:10,920
makes sense.

504
00:33:10,920 --> 00:33:13,250
I gave you one argument
last time.

505
00:33:13,250 --> 00:33:14,930
I'll give you another
argument this time.

506
00:33:18,780 --> 00:33:28,730
If S sub J is greater than or
equal to 0, we're going to

507
00:33:28,730 --> 00:33:32,550
decide that 0 is the
correct hypothesis.

508
00:33:35,300 --> 00:33:40,370
If H equals 1 is the correct
hypothesis, then we're going

509
00:33:40,370 --> 00:33:42,750
make an error when S
sub J is greater

510
00:33:42,750 --> 00:33:44,571
than or equal to alpha.

511
00:33:44,571 --> 00:33:47,010
If S sub J is less than or equal
to the beta, we're going

512
00:33:47,010 --> 00:33:50,410
to decide H equals 1.

513
00:33:50,410 --> 00:33:54,810
And conditional on H equals 1,
an error is made if SJ is

514
00:33:54,810 --> 00:33:56,560
greater than or equal
to alpha.

515
00:33:56,560 --> 00:34:00,710
Conditional on H equals 0, an
error is made if SJ is less

516
00:34:00,710 --> 00:34:02,570
than or equal to beta.

517
00:34:02,570 --> 00:34:06,880
OK, so the probability of the
error conditional on H equals

518
00:34:06,880 --> 00:34:12,090
1 is the probability that S sub
J is greater than or equal

519
00:34:12,090 --> 00:34:16,370
to alpha, given H equals 1,
which is less than or equal to

520
00:34:16,370 --> 00:34:18,120
E to the minus alpha or star.

521
00:34:18,120 --> 00:34:22,340
This is the thing that
we said before.

522
00:34:22,340 --> 00:34:25,270
r star is the root
of gamma of r.

523
00:34:25,270 --> 00:34:29,840
And gamma of r is
equal to this.

524
00:34:29,840 --> 00:34:38,060
OK, so, let's just make life a
little easier for ourselves

525
00:34:38,060 --> 00:34:43,050
assume that our a priori
probabilities are each 1/2.

526
00:34:43,050 --> 00:34:47,810
This is also called maximum
likelihood decision.

527
00:34:47,810 --> 00:34:51,300
You take this likelihood ratio,
and you just decide on

528
00:34:51,300 --> 00:34:54,770
the basis of the likelihood
ratio.

529
00:34:54,770 --> 00:34:58,790
OK, then at the end of trial
end, the probability of H

530
00:34:58,790 --> 00:35:03,850
equals 0 given Sn divided by the
probability of H equals 1

531
00:35:03,850 --> 00:35:07,120
given Sn, the a prioris
cancel out.

532
00:35:07,120 --> 00:35:11,260
It is just E to the S sub n.

533
00:35:11,260 --> 00:35:12,510
That's what it is.

534
00:35:15,000 --> 00:35:17,230
It's the likelihood ratio.

535
00:35:17,230 --> 00:35:21,100
S sub n is the log-likelihood
ratio.

536
00:35:21,100 --> 00:35:24,360
So this is what it is.

537
00:35:24,360 --> 00:35:30,190
If you now take probability of H
equals 0 on probability that

538
00:35:30,190 --> 00:35:34,130
H equals 1 given S of
in, this equation

539
00:35:34,130 --> 00:35:36,440
becomes this equation.

540
00:35:36,440 --> 00:35:40,560
And then the probability of H
equals 1 given S sub n is just

541
00:35:40,560 --> 00:35:44,910
E to the minus Sn over 1
plus E to the minus Sn.

542
00:35:44,910 --> 00:35:50,210
Now if Sn is a large number, E
to the minus Sn is going to be

543
00:35:50,210 --> 00:35:52,120
totally trivial.

544
00:35:52,120 --> 00:35:55,570
And the probability that
H equals 1 given Sn is

545
00:35:55,570 --> 00:35:58,730
essentially E to the minus Sn.

546
00:35:58,730 --> 00:36:04,260
It means when you can choose
different values of n, this

547
00:36:04,260 --> 00:36:06,990
very directly gives you
a control on what the

548
00:36:06,990 --> 00:36:08,900
probability of error is.

549
00:36:08,900 --> 00:36:13,680
The probability of error is
essentially E to the minus Sn.

550
00:36:13,680 --> 00:36:17,210
So if you choose a threshold
alpha, what you're doing is

551
00:36:17,210 --> 00:36:20,450
you're guaranteeing that the
probability of error cannot be

552
00:36:20,450 --> 00:36:23,080
less than E to the
minus alpha.

553
00:36:23,080 --> 00:36:27,650
OK, so this is more than just
talking about averages.

554
00:36:27,650 --> 00:36:31,120
This is saying if you use a
threshold rule, then what

555
00:36:31,120 --> 00:36:34,600
you're doing is guaranteeing
that the probability of error

556
00:36:34,600 --> 00:36:37,360
is never going to be less
than this quantity

557
00:36:37,360 --> 00:36:39,720
of specified here.

558
00:36:39,720 --> 00:36:43,940
OK, we saw last time the cost of
choosing alpha to be large

559
00:36:43,940 --> 00:36:48,060
is that you have to make a very
large number of trials,

560
00:36:48,060 --> 00:36:50,010
at least given H equals 0.

561
00:36:50,010 --> 00:36:53,770
Why don't I worry about
the number of trials

562
00:36:53,770 --> 00:36:56,840
for H equals 1?

563
00:36:56,840 --> 00:37:00,590
I mean, it's nothing to be
thought through here.

564
00:37:00,590 --> 00:37:04,370
If my thresholds are large,
my probability of

565
00:37:04,370 --> 00:37:05,620
error is very small.

566
00:37:08,060 --> 00:37:17,530
The expected values of things
for very large log-likelihood

567
00:37:17,530 --> 00:37:22,800
ratios are determined almost
entirely by H equals 0.

568
00:37:22,800 --> 00:37:24,610
H equals 1 sometimes.

569
00:37:24,610 --> 00:37:27,250
You sometimes make a mistake,
because it's something very,

570
00:37:27,250 --> 00:37:28,730
very unusual.

571
00:37:28,730 --> 00:37:31,450
But that has very little
influence on the expected

572
00:37:31,450 --> 00:37:33,680
number of tests you're making.

573
00:37:33,680 --> 00:37:38,960
So what happens then is the
expected number of tests you

574
00:37:38,960 --> 00:37:42,790
make under the hypothesis
that H is equal to 0--

575
00:37:45,580 --> 00:37:47,440
now we're using Wald's equality

576
00:37:47,440 --> 00:37:49,540
rather than Wald's identity--

577
00:37:49,540 --> 00:37:52,720
it's equal to the expected
value of S sub J given H

578
00:37:52,720 --> 00:37:56,530
equals 0, divided by the
expected value of Z,

579
00:37:56,530 --> 00:37:57,780
given H equals 0.

580
00:38:01,150 --> 00:38:05,940
Z is the log-likelihood
ratio of one trial.

581
00:38:05,940 --> 00:38:10,650
This is just Wald's
equality with this

582
00:38:10,650 --> 00:38:12,710
condition thrown into it.

583
00:38:12,710 --> 00:38:17,290
Now what's the expected value
of SJ given H equals 0?

584
00:38:17,290 --> 00:38:20,780
It's essentially alpha, and if
you want to be more careful,

585
00:38:20,780 --> 00:38:26,720
it's alpha plus the expected
overshoot given H equals 0.

586
00:38:26,720 --> 00:38:29,960
And that's divided by the
expected value of Z,

587
00:38:29,960 --> 00:38:31,090
given H equals 0.

588
00:38:31,090 --> 00:38:32,990
This is the answer
we got last time.

589
00:38:36,260 --> 00:38:39,520
So the number of tests you have
to make, if you set a

590
00:38:39,520 --> 00:38:43,820
positive threshold alpha, is
essentially the number of

591
00:38:43,820 --> 00:38:48,310
tests you have to make when
the hypothesis is equal 0.

592
00:38:48,310 --> 00:38:52,400
So the funny thing which is
happening here is that as you

593
00:38:52,400 --> 00:38:56,760
change alpha, you're changing
the probability of error for

594
00:38:56,760 --> 00:39:00,810
hypothesis H equals 1.

595
00:39:00,810 --> 00:39:02,830
And you're changing the number
of tests you're going to have

596
00:39:02,830 --> 00:39:05,290
to do when H is equal to 0.

597
00:39:05,290 --> 00:39:08,650
When you change beta, it's
just the opposite.

598
00:39:08,650 --> 00:39:13,130
So that when you change beta, if
you make beta a very large

599
00:39:13,130 --> 00:39:17,720
negative, you have to make an
enormous number of tests under

600
00:39:17,720 --> 00:39:20,750
the circumstance that
H is equal to 1.

601
00:39:20,750 --> 00:39:24,700
But you might make an error
when H is equal to 0.

602
00:39:24,700 --> 00:39:30,270
So the trade off is between
number of trials under one

603
00:39:30,270 --> 00:39:33,700
hypothesis, error of probability
under the other

604
00:39:33,700 --> 00:39:34,950
hypothesis.

605
00:39:39,300 --> 00:39:43,140
That's almost all we wanted to
say about Wald's identity.

606
00:39:43,140 --> 00:39:46,426
There's one other huge thing
that we want to talk about.

607
00:39:48,990 --> 00:39:54,470
If you take the first two
derivatives of Wald's identity

608
00:39:54,470 --> 00:40:00,360
at r equals 0, you get some
interesting things coming out.

609
00:40:00,360 --> 00:40:03,080
I mean, Wald's identity, you
can use it any value

610
00:40:03,080 --> 00:40:05,430
of r you want to.

611
00:40:05,430 --> 00:40:08,190
And when you use it for a large
value of r, you that an

612
00:40:08,190 --> 00:40:11,520
interesting result about
large deviations.

613
00:40:11,520 --> 00:40:15,650
When you use it at a small value
of r, you get something

614
00:40:15,650 --> 00:40:18,220
more about typical cases.

615
00:40:18,220 --> 00:40:25,590
So looking at it at r equals 0,
what you want to do is you

616
00:40:25,590 --> 00:40:28,970
want to take the derivative
with respect

617
00:40:28,970 --> 00:40:34,170
to r of Wald's identity.

618
00:40:34,170 --> 00:40:37,720
This expected value in here
we know is equal to 1.

619
00:40:37,720 --> 00:40:41,050
It's equal to 1 whatever
value of r we choose.

620
00:40:41,050 --> 00:40:45,140
And therefore, when we take
the derivative of this, we

621
00:40:45,140 --> 00:40:46,890
have to get 0.

622
00:40:46,890 --> 00:40:49,150
But we also want to take
the derivative of it

623
00:40:49,150 --> 00:40:50,670
to see what we get.

624
00:40:50,670 --> 00:40:53,460
So when you take the derivative
of this quantity

625
00:40:53,460 --> 00:40:56,260
here and you don't worry
about what exists and

626
00:40:56,260 --> 00:40:57,510
what doesn't exist--

627
00:41:01,250 --> 00:41:03,290
you have to take the
derivative here--

628
00:41:03,290 --> 00:41:07,470
so you get an S sub J there.

629
00:41:07,470 --> 00:41:09,340
You take the derivative
here, you get a

630
00:41:09,340 --> 00:41:11,190
gamma prime of r there.

631
00:41:11,190 --> 00:41:15,440
If you get SJ minus J times
gamma prime of r, and this E

632
00:41:15,440 --> 00:41:19,110
to the what have you
just sits there.

633
00:41:19,110 --> 00:41:22,450
You take the derivative of E
to something, you never get

634
00:41:22,450 --> 00:41:23,580
rid of the E to something.

635
00:41:23,580 --> 00:41:27,300
You just get piled up stuff
in front of it.

636
00:41:27,300 --> 00:41:33,840
OK, so when we evaluate that at
r equals 0, what happens?

637
00:41:33,840 --> 00:41:36,380
Well, what's the value of
the gamma prime of 0?

638
00:41:40,180 --> 00:41:42,120
It's the expected
value of X, yes.

639
00:41:48,060 --> 00:41:53,860
And this quantity here is all
equal to 1, so we can forget

640
00:41:53,860 --> 00:41:56,140
about that.

641
00:41:56,140 --> 00:42:00,530
When r is equal to 0,
this is equal to 0.

642
00:42:00,530 --> 00:42:04,320
When r is equal to 0, gamma
of r is equal to 0.

643
00:42:04,320 --> 00:42:07,810
So this whole thing
in here is 0.

644
00:42:07,810 --> 00:42:09,510
So E to the 0 is 1.

645
00:42:09,510 --> 00:42:11,070
So we've got a 1 there.

646
00:42:11,070 --> 00:42:16,870
We got expected value of
S sub J minus J times X

647
00:42:16,870 --> 00:42:19,770
bar is equal to 0.

648
00:42:19,770 --> 00:42:20,290
What is that?

649
00:42:20,290 --> 00:42:22,190
That's Wald's equality.

650
00:42:22,190 --> 00:42:27,260
So Wald's equality falls out of
the Wald's identity as what

651
00:42:27,260 --> 00:42:31,630
happens as the derivative
of Wald's identity

652
00:42:31,630 --> 00:42:34,000
that r equals 0.

653
00:42:34,000 --> 00:42:36,975
Well, since we're so successful
with that, let's go

654
00:42:36,975 --> 00:42:38,360
on and take another
derivative.

655
00:42:40,860 --> 00:42:41,566
Yes?

656
00:42:41,566 --> 00:42:46,180
AUDIENCE: I guess you want the
final equal to 0 [INAUDIBLE].

657
00:42:46,180 --> 00:42:49,970
PROFESSOR: Oh, the final equal
to 0 comes from the fact that

658
00:42:49,970 --> 00:42:54,080
this quantity here that you're
starting with is equal to 1

659
00:42:54,080 --> 00:42:56,120
for all values of r.

660
00:42:56,120 --> 00:42:57,680
Therefore, I want to take
the derivative with

661
00:42:57,680 --> 00:43:01,130
respect to r, I get 0.

662
00:43:01,130 --> 00:43:02,750
So that's one equation.

663
00:43:02,750 --> 00:43:05,100
The other thing is I just go
through the mechanics of

664
00:43:05,100 --> 00:43:07,400
taking the derivative.

665
00:43:07,400 --> 00:43:11,720
OK, so let's try to take
the second derivative.

666
00:43:11,720 --> 00:43:16,360
Take the second derivative by
taking the derivative of the

667
00:43:16,360 --> 00:43:18,290
first derivative.

668
00:43:18,290 --> 00:43:26,370
And what happens is then is this
quantity in here I get an

669
00:43:26,370 --> 00:43:29,320
extra term of that sitting
over there.

670
00:43:29,320 --> 00:43:31,840
And along with that, I get the
derivative of this with

671
00:43:31,840 --> 00:43:33,090
respect to r.

672
00:43:35,480 --> 00:43:39,060
I should probably have written
that down there but since I

673
00:43:39,060 --> 00:43:47,160
didn't, let me see
if I can do it.

674
00:43:47,160 --> 00:44:03,740
I get the expected value of SJ
minus J gamma prime of r.

675
00:44:03,740 --> 00:44:05,650
And this quantity
is squared now,

676
00:44:05,650 --> 00:44:07,450
because I have this there.

677
00:44:07,450 --> 00:44:11,780
I'm taking the derivative of
this term with respect to r.

678
00:44:11,780 --> 00:44:16,190
And also, I have to take the
derivative of this with

679
00:44:16,190 --> 00:44:17,700
respect to r.

680
00:44:17,700 --> 00:44:25,630
So that gives me minus J times
gamma double prime of r.

681
00:44:25,630 --> 00:44:41,570
And all of this times E to the
r SJ minus J gamma of r.

682
00:44:45,260 --> 00:44:49,990
Now I want to evaluate
this at r equals 0.

683
00:44:49,990 --> 00:44:54,510
Evaluating this at r equals
0, this term goes away.

684
00:44:54,510 --> 00:45:04,030
So I wind up with the expected
value of SJ minus J gamma

685
00:45:04,030 --> 00:45:12,240
prime of r where minus J
gamma double prime of

686
00:45:12,240 --> 00:45:17,740
r is equal to 0.

687
00:45:17,740 --> 00:45:20,060
Well, this doesn't look bad.

688
00:45:20,060 --> 00:45:23,700
But if you try to use it if you
expand this term here, you

689
00:45:23,700 --> 00:45:30,480
get a term the expected value of
S of J times J. And you can

690
00:45:30,480 --> 00:45:32,440
struggle with that.

691
00:45:32,440 --> 00:45:33,900
And it's ugly.

692
00:45:33,900 --> 00:45:34,920
That's very ugly.

693
00:45:34,920 --> 00:45:43,340
But if you now say, if
we have a mean we

694
00:45:43,340 --> 00:45:45,200
can use Wald's equality.

695
00:45:45,200 --> 00:45:47,150
It tells us what we
want to know.

696
00:45:47,150 --> 00:45:51,170
If we don't have a mean, then
Wald's equality doesn't tell

697
00:45:51,170 --> 00:45:53,320
us anything.

698
00:45:53,320 --> 00:45:55,670
But this is going to
tell us something.

699
00:45:55,670 --> 00:45:59,100
So we're going to make the
assumption here that r is

700
00:45:59,100 --> 00:46:06,525
equal to 0 and X bar
is equal to 0.

701
00:46:09,340 --> 00:46:13,590
And if X bar is equal
to 0, gamma prime of

702
00:46:13,590 --> 00:46:17,170
0 is equal to 0.

703
00:46:17,170 --> 00:46:29,540
And gamma double prime of 0 is
equal to sigma squared of X.

704
00:46:29,540 --> 00:46:31,810
So you do all of that.

705
00:46:31,810 --> 00:46:37,030
What you get is the expected
value of S sub J squared minus

706
00:46:37,030 --> 00:46:42,070
sigma X squared of
J is equal to 0.

707
00:46:42,070 --> 00:46:45,460
This is the same kind
of thing that we

708
00:46:45,460 --> 00:46:48,360
got from Wald's equality.

709
00:46:48,360 --> 00:46:52,130
From Wald's equality, it didn't
tell us anything.

710
00:46:52,130 --> 00:46:56,100
It just gave us a relationship
between the expected value of

711
00:46:56,100 --> 00:46:59,780
S sub J and expected
value of J. This is

712
00:46:59,780 --> 00:47:01,180
doing the same thing.

713
00:47:01,180 --> 00:47:05,540
It's giving us a relationship
between the expected value of

714
00:47:05,540 --> 00:47:12,040
S sub J squared and the expected
value of J. So we get

715
00:47:12,040 --> 00:47:13,930
the same kind of quantity.

716
00:47:13,930 --> 00:47:16,970
It's doing the same
thing for us.

717
00:47:16,970 --> 00:47:27,530
Now you look at this for a 0
means simple random walk.

718
00:47:27,530 --> 00:47:29,590
Now you would have thought
before you started to take

719
00:47:29,590 --> 00:47:34,390
this class that a simple random
walk with mean 0 was

720
00:47:34,390 --> 00:47:37,350
the simplest thing
in the world.

721
00:47:37,350 --> 00:47:41,010
And we've seen by looking at
that it really isn't all that

722
00:47:41,010 --> 00:47:45,830
simple, that you come play these
silly games like you can

723
00:47:45,830 --> 00:47:50,750
gamble forever, where with
probability 1/2 you lose $1,

724
00:47:50,750 --> 00:47:53,590
with probability 1/2
you win $1--

725
00:47:53,590 --> 00:47:55,420
perfectly fair game.

726
00:47:55,420 --> 00:47:58,365
And with probability one, you
to make $1 out of that, and

727
00:47:58,365 --> 00:48:00,140
quit and go home.

728
00:48:00,140 --> 00:48:03,470
And since you to make $1 out of
it, and quit and go home,

729
00:48:03,470 --> 00:48:06,610
you can then quickly come
back again and it again.

730
00:48:06,610 --> 00:48:07,960
You can make $2.

731
00:48:07,960 --> 00:48:09,440
You can make $10.

732
00:48:09,440 --> 00:48:10,750
You can make $1,000.

733
00:48:10,750 --> 00:48:14,770
You can make $1 million
with probability 1.

734
00:48:14,770 --> 00:48:17,840
So the simple random walk
is no longer simple.

735
00:48:17,840 --> 00:48:19,820
It becomes puzzling.

736
00:48:19,820 --> 00:48:23,910
But Wald's identity is dealing
with two thresholds, one of

737
00:48:23,910 --> 00:48:27,390
alpha and one at beta.

738
00:48:27,390 --> 00:48:34,380
When we apply this and you
observe it as a simple random

739
00:48:34,380 --> 00:48:39,810
walk, where you either go up
by 1 or go down by 1, each

740
00:48:39,810 --> 00:48:44,410
with probability 1/2, the
mean of X is 0 and the

741
00:48:44,410 --> 00:48:46,730
variance of X is 1.

742
00:48:46,730 --> 00:48:49,900
So this quantity here is 1.

743
00:48:49,900 --> 00:48:56,240
You can then play games with
what the probability is that

744
00:48:56,240 --> 00:48:59,526
you hit the upper threshold and
the probability that you

745
00:48:59,526 --> 00:49:01,150
hit the lower threshold.

746
00:49:01,150 --> 00:49:02,270
I mean, it's done in the text.

747
00:49:02,270 --> 00:49:05,390
You don't have to take
my word for it.

748
00:49:05,390 --> 00:49:09,680
And when you do that, what you
find is the expected value of

749
00:49:09,680 --> 00:49:14,230
J is equal to minus
beta times alpha.

750
00:49:14,230 --> 00:49:18,000
Theta is a negative number,
remember, so this is expected

751
00:49:18,000 --> 00:49:22,690
value of J is the magnitude
of theta times the

752
00:49:22,690 --> 00:49:23,940
magnitude of alpha.

753
00:49:26,290 --> 00:49:30,230
Now that's a little bizarre, but
then you think about it a

754
00:49:30,230 --> 00:49:31,370
little bit.

755
00:49:31,370 --> 00:49:34,410
You think what happens.

756
00:49:34,410 --> 00:49:36,310
And this is really exact.

757
00:49:36,310 --> 00:49:40,720
I mean, this isn't an
approximation or anything.

758
00:49:40,720 --> 00:49:45,000
If alpha is very large, and
beta is very large and

759
00:49:45,000 --> 00:49:50,050
negative, and you play this
random walk game, you're going

760
00:49:50,050 --> 00:49:52,250
to fluctuate a long time.

761
00:49:52,250 --> 00:49:54,350
You're going to disperse
slowly.

762
00:49:54,350 --> 00:49:56,920
You're going to disperse
according to the square root

763
00:49:56,920 --> 00:50:00,010
of n, or the number
of tests you take.

764
00:50:00,010 --> 00:50:03,720
So the amount of time it takes
you until you get way out to

765
00:50:03,720 --> 00:50:08,330
these thresholds should be--

766
00:50:08,330 --> 00:50:16,080
to the namely the value
that n has to have--

767
00:50:16,080 --> 00:50:19,990
roughly the square of alpha
when beta and alpha

768
00:50:19,990 --> 00:50:21,040
are both the same.

769
00:50:21,040 --> 00:50:22,820
This is something more
general than that.

770
00:50:22,820 --> 00:50:27,950
It says that if Sn, the
stop-when-you're-ahead game,

771
00:50:27,950 --> 00:50:34,820
we make alpha equals 1, the
expected value of J depends on

772
00:50:34,820 --> 00:50:37,110
what the lower threshold is.

773
00:50:37,110 --> 00:50:41,870
And that suddenly makes sense,
because what that's saying is

774
00:50:41,870 --> 00:50:45,530
if we have a lower threshold at
10, an upper threshold at

775
00:50:45,530 --> 00:50:54,170
one, then most of the
time you win.

776
00:50:54,170 --> 00:50:57,250
But when you lose,
you lose $10.

777
00:50:57,250 --> 00:50:58,900
When you win, you win, $1.

778
00:50:58,900 --> 00:51:02,000
When you set a lower threshold
at 100, when you

779
00:51:02,000 --> 00:51:03,800
lose, you lose $100.

780
00:51:03,800 --> 00:51:06,370
When you win, you win $1.

781
00:51:06,370 --> 00:51:09,310
And suddenly, that
stop-when-you're-ahead game

782
00:51:09,310 --> 00:51:13,300
does not look quite as
attractive as it did before.

783
00:51:13,300 --> 00:51:17,290
What you're doing is taking a
chance where you're probably

784
00:51:17,290 --> 00:51:22,950
going to win of winning $1, and
you're risking your life's

785
00:51:22,950 --> 00:51:27,470
assets for it, which doesn't
make too much sense anymore.

786
00:51:27,470 --> 00:51:32,200
OK this, I think, it gives you
a better idea of what's going

787
00:51:32,200 --> 00:51:36,600
on on the simple random walk
than anything else I've seen.

788
00:51:50,210 --> 00:51:53,100
So it's time to start talking
about martingales.

789
00:51:56,570 --> 00:51:59,740
A martingale, like most of the
other things we've been

790
00:51:59,740 --> 00:52:02,120
talking about in the
course, is a

791
00:52:02,120 --> 00:52:05,320
sequence of random variables.

792
00:52:05,320 --> 00:52:08,650
This is a more general kind of
sequence than most of them.

793
00:52:12,085 --> 00:52:16,200
Almost all of the processes
we've talked about so far have

794
00:52:16,200 --> 00:52:20,400
been the kinds of things you can
sort of get your hands on.

795
00:52:20,400 --> 00:52:23,950
And this is defined very
abstractly in terms of a

796
00:52:23,950 --> 00:52:26,690
peculiar property that it has.

797
00:52:26,690 --> 00:52:31,500
And then the peculiar property
it has is the expected value

798
00:52:31,500 --> 00:52:36,320
as the nth term, in this thing
called a martingale,

799
00:52:36,320 --> 00:52:40,010
conditional on knowing the
values of all the previous

800
00:52:40,010 --> 00:52:45,340
values, expected value of Z sub
n given the value of Z and

801
00:52:45,340 --> 00:52:50,130
minus 1, Z and minus 2, all the
way down to Z1 is equal to

802
00:52:50,130 --> 00:52:52,590
Z sub n minus 1.

803
00:52:52,590 --> 00:53:00,330
Namely, the expected value here
is what you had there.

804
00:53:00,330 --> 00:53:05,370
The word martingale comes from
gambling, where gamblers used

805
00:53:05,370 --> 00:53:09,740
to spend a great deal of time
trying to find gambling

806
00:53:09,740 --> 00:53:15,020
strategies when to stop, when to
start betting bigger, when

807
00:53:15,020 --> 00:53:18,460
to start betting smaller, when
to do all sorts of things, all

808
00:53:18,460 --> 00:53:23,050
sorts of strategies for how
to lose less money.

809
00:53:23,050 --> 00:53:27,520
Let me put it that way, because
you rarely find that

810
00:53:27,520 --> 00:53:30,890
opportunity where you can
play a fair game.

811
00:53:30,890 --> 00:53:37,450
But if you play a fair game,
martingales are what sort of

812
00:53:37,450 --> 00:53:38,650
rules on that.

813
00:53:38,650 --> 00:53:42,560
And what that says is if you
play this game for a long

814
00:53:42,560 --> 00:53:47,480
time, your capital is
Z sub n minus 1.

815
00:53:50,790 --> 00:53:54,090
This says, figure expected
capital after you

816
00:53:54,090 --> 00:53:56,790
play one more time.

817
00:53:56,790 --> 00:54:01,640
No matter what strategy you use,
your expected capital is

818
00:54:01,640 --> 00:54:05,460
going to be the same as
was as the actual

819
00:54:05,460 --> 00:54:09,170
capital the time before.

820
00:54:09,170 --> 00:54:14,360
If this is too abstract for you,
and it's too abstract for

821
00:54:14,360 --> 00:54:18,010
me half the time, because I look
at this, and I say, gee,

822
00:54:18,010 --> 00:54:20,840
that's not much of a
restriction, is it?

823
00:54:20,840 --> 00:54:24,380
What we're talking about is
expected values here.

824
00:54:24,380 --> 00:54:28,810
But it's more than that, because
it's saying for every

825
00:54:28,810 --> 00:54:33,990
choice of sample value for all
of these things, none of them

826
00:54:33,990 --> 00:54:37,440
make any difference, except
the last one.

827
00:54:37,440 --> 00:54:39,970
And that's what happens
in gambling.

828
00:54:39,970 --> 00:54:44,130
It doesn't make any difference
how your capital has gotten to

829
00:54:44,130 --> 00:54:49,310
the point where it is
at time n minus 1.

830
00:54:49,310 --> 00:54:55,880
You make a bet in a fair bet,
and what you win is solely a

831
00:54:55,880 --> 00:55:01,020
function of what you've bet,
if the game is fair.

832
00:55:01,020 --> 00:55:02,270
And that's what this
is saying.

833
00:55:05,790 --> 00:55:09,510
So when you write it out this
way, the expected value of Zn,

834
00:55:09,510 --> 00:55:13,820
given that 1, the random
variable Zn minus 1 has a

835
00:55:13,820 --> 00:55:16,900
particular value Zn minus 1.

836
00:55:19,920 --> 00:55:24,530
You started out with
a particular value.

837
00:55:24,530 --> 00:55:29,520
It says that expected value
is equal to what you

838
00:55:29,520 --> 00:55:32,870
said at the last time.

839
00:55:32,870 --> 00:55:36,170
And this is true for all sample
values Zn minus 1 down

840
00:55:36,170 --> 00:55:40,080
to Z1, which is why it's a much
stronger statement than

841
00:55:40,080 --> 00:55:41,330
it appears to be.

842
00:55:43,910 --> 00:55:46,200
now there's a lemma.

843
00:55:46,200 --> 00:55:49,250
I want to talk about that a
little bit, because it's a

844
00:55:49,250 --> 00:55:51,500
good time to get you
used to what these

845
00:55:51,500 --> 00:55:53,055
expected values mean.

846
00:55:55,810 --> 00:56:02,040
For martingale, the expected
value of Zn given Zi, Zi minus

847
00:56:02,040 --> 00:56:04,080
1, all the way down to Z1.

848
00:56:04,080 --> 00:56:07,770
This expected value is
equal to Z sub i.

849
00:56:07,770 --> 00:56:12,560
In other words, it's not only
that your expected capital,

850
00:56:12,560 --> 00:56:18,030
given all of the past, is equal
to what you had on the

851
00:56:18,030 --> 00:56:19,880
last time instant.

852
00:56:19,880 --> 00:56:25,830
If you're not given anything
for 100 years back, and all

853
00:56:25,830 --> 00:56:33,750
you know is what your capital
was 100 years ago, and if we

854
00:56:33,750 --> 00:56:36,180
think we're playing a fair game
all of this time, which

855
00:56:36,180 --> 00:56:39,840
is of course always a question,
the expected value

856
00:56:39,840 --> 00:56:45,530
of what we have now, conditional
on everything from

857
00:56:45,530 --> 00:56:51,730
100 years back through
recorded history, is

858
00:56:51,730 --> 00:56:53,400
just that last term.

859
00:56:53,400 --> 00:56:58,350
In other words, it's the same
kind of isolation of the past

860
00:56:58,350 --> 00:57:01,790
from the future as we had
with Markov change.

861
00:57:01,790 --> 00:57:05,260
With Markov change, remember,
it's only what happens at one

862
00:57:05,260 --> 00:57:10,990
instant, given what happens a
one instant, it makes the past

863
00:57:10,990 --> 00:57:13,080
independent of the future.

864
00:57:13,080 --> 00:57:17,430
Here it's not quite that way,
because the past and the

865
00:57:17,430 --> 00:57:22,140
future are separated only in
terms of the details of the

866
00:57:22,140 --> 00:57:27,135
past, and the expected
value the future.

867
00:57:27,135 --> 00:57:33,320
It says expected value of Z sub
n given all of the details

868
00:57:33,320 --> 00:57:36,640
of the past, no matter what the
details of the past are,

869
00:57:36,640 --> 00:57:41,340
the effective value of Zn is
equal to the actual value at

870
00:57:41,340 --> 00:57:43,440
times Z sub i.

871
00:57:43,440 --> 00:57:48,780
So I want to improve this for
you, and I warn you you're not

872
00:57:48,780 --> 00:57:50,880
going to follow this proof.

873
00:57:50,880 --> 00:57:53,820
And that's part of the reason
for me to do it, because I

874
00:57:53,820 --> 00:57:59,840
want you to go back to Chapter
1 and think that through.

875
00:57:59,840 --> 00:58:02,740
Because in dealing with
martingales, you have to think

876
00:58:02,740 --> 00:58:04,300
this through.

877
00:58:04,300 --> 00:58:07,240
Because if you don't think it
through, you're stuck with

878
00:58:07,240 --> 00:58:10,760
this notation all
the way through.

879
00:58:10,760 --> 00:58:14,730
And if you try to use this
notation with martingales,

880
00:58:14,730 --> 00:58:18,330
this is nice notation when you
get confused, but you don't

881
00:58:18,330 --> 00:58:19,950
want to use it all the time.

882
00:58:19,950 --> 00:58:23,500
So you have to be able to go
through arguments like this.

883
00:58:23,500 --> 00:58:30,860
What I want to show is that if
E to the Z3, given Z1 and Z2

884
00:58:30,860 --> 00:58:37,390
is equal to Z2, then one special
case of this lemma is

885
00:58:37,390 --> 00:58:45,370
that expected value of Z3
given Z1 is equal to Z1.

886
00:58:45,370 --> 00:58:47,480
And how do we show that?

887
00:58:47,480 --> 00:58:51,930
Well, what we do is we use this
law complete expectation.

888
00:58:55,930 --> 00:59:00,500
Well, first we remember the
expected value of an arbitrary

889
00:59:00,500 --> 00:59:06,830
random variable X is the
expected value of the expected

890
00:59:06,830 --> 00:59:10,050
value of X given Y. Now
what does that mean?

891
00:59:10,050 --> 00:59:17,000
The expected value of the random
variable X, given Y, is

892
00:59:17,000 --> 00:59:18,990
a random variable.

893
00:59:18,990 --> 00:59:22,500
It's a random variable which
depends on Y. That's a

894
00:59:22,500 --> 00:59:27,250
function of the sample value of
Y. Namely, if you look at

895
00:59:27,250 --> 00:59:30,990
this quantity up here,
expected value of X

896
00:59:30,990 --> 00:59:33,290
given Y equals 1.

897
00:59:33,290 --> 00:59:36,130
Expected value of X
given Y equals 2.

898
00:59:36,130 --> 00:59:38,830
Expected value of X
given Y equals 3.

899
00:59:38,830 --> 00:59:41,010
We have all of these
values here.

900
00:59:41,010 --> 00:59:43,720
We have a probability
measure on it.

901
00:59:43,720 --> 00:59:51,150
This is a random variable,
which is a function of Y.

902
00:59:51,150 --> 00:59:54,710
You've averaged that over X,
but you're left move why

903
00:59:54,710 --> 00:59:56,810
because of the conditioning
here.

904
00:59:56,810 --> 01:00:04,440
So this quantity in here is now
a function of Y. So when

905
01:00:04,440 --> 01:00:11,290
we take this equation and we
add the conditioning on Z1,

906
01:00:11,290 --> 01:00:16,920
namely, this is being
used for Z3 and Z2.

907
01:00:16,920 --> 01:00:26,910
Expected value of Z3 is equal to
the expected value over Z2

908
01:00:26,910 --> 01:00:32,340
of the expected value of Z3
given Z2, whole thing

909
01:00:32,340 --> 01:00:35,180
dependent on Z1.

910
01:00:35,180 --> 01:00:39,040
OK, so what it says is this
expected value is the expected

911
01:00:39,040 --> 01:00:46,110
value of the expected value of
Z3 condition on Z2 and Z1.

912
01:00:46,110 --> 01:00:50,840
This quantity here as
a function of what?

913
01:00:50,840 --> 01:00:51,990
That's a random variable.

914
01:00:51,990 --> 01:00:55,136
It's a function of what
random variables?

915
01:00:55,136 --> 01:00:56,540
AUDIENCE: Z2, Z1.

916
01:00:56,540 --> 01:00:59,350
PROFESSOR: Z1 and Z2, yes.

917
01:00:59,350 --> 01:01:02,380
So this is a function
of Z1 and Z2.

918
01:01:02,380 --> 01:01:06,220
What value is it as a function
of Z1 and Z2?

919
01:01:06,220 --> 01:01:09,040
It's just equal to Z2.

920
01:01:09,040 --> 01:01:12,360
So this quantity
in here is Z2.

921
01:01:12,360 --> 01:01:17,620
so we're asking what's expected
value of Z2 given Z1.

922
01:01:17,620 --> 01:01:23,960
And by definition of martingale,
it is equal to Z1.

923
01:01:27,130 --> 01:01:30,910
Now I imagine about half you
could follow that, and half of

924
01:01:30,910 --> 01:01:35,460
you couldn't, and half of
you sort of followed it.

925
01:01:35,460 --> 01:01:39,530
This is a kind of argument we'll
be using all the way

926
01:01:39,530 --> 01:01:41,840
through on this stuff.

927
01:01:41,840 --> 01:01:43,310
So make sure you
understand it.

928
01:01:45,990 --> 01:01:49,220
I mean, once you get
it, it's easy.

929
01:01:49,220 --> 01:01:52,390
And you can apply it in
all sorts of places.

930
01:01:52,390 --> 01:01:55,010
So it's worth doing it.

931
01:01:55,010 --> 01:01:57,650
In the same way, you can
follow the same kind of

932
01:01:57,650 --> 01:02:02,470
argument through the expected
value Z sub i plus 2, using

933
01:02:02,470 --> 01:02:09,020
this total expectation
based on Zi plus 1.

934
01:02:09,020 --> 01:02:11,900
And you go through
the whole thing.

935
01:02:11,900 --> 01:02:17,770
When you go down to i equals 1,
it says the expected value

936
01:02:17,770 --> 01:02:22,330
of z is equal to the expected
value of Z1.

937
01:02:22,330 --> 01:02:26,420
If you want to become wealthy,
have a wealthy parent, who

938
01:02:26,420 --> 01:02:28,880
leads to a lot of money
20 years ago.

939
01:02:28,880 --> 01:02:34,600
That's the easiest way to make
a million dollars is to start

940
01:02:34,600 --> 01:02:37,580
out with 2 million dollars
is the way that

941
01:02:37,580 --> 01:02:40,830
some people put it.

942
01:02:40,830 --> 01:02:45,770
OK, let's have some simple
examples a martingales.

943
01:02:45,770 --> 01:02:49,536
One of them is a zero-mean
random walk.

944
01:02:49,536 --> 01:02:52,370
Mainly, what I'm trying to do
here is to show you this

945
01:02:52,370 --> 01:02:56,320
martingales are really pretty
general things.

946
01:02:56,320 --> 01:03:00,480
And since there are many very
general theorems that hold for

947
01:03:00,480 --> 01:03:03,980
all martingales, you can then
apply them to all of these

948
01:03:03,980 --> 01:03:07,640
special cases, which
is kind of neat.

949
01:03:07,640 --> 01:03:14,530
To have a zero-mean random walk,
let Z sub n be the sum

950
01:03:14,530 --> 01:03:23,720
of X1 plus Xn and the X sub
i's are IID and zero mean.

951
01:03:23,720 --> 01:03:26,710
The fact that they're IID
makes it a random walk.

952
01:03:26,710 --> 01:03:30,060
The fact that there's zero mean
makes it a special zero

953
01:03:30,060 --> 01:03:34,790
mean random walk, and the
expected value of Z sub n,

954
01:03:34,790 --> 01:03:38,860
given Zn minus 1.

955
01:03:38,860 --> 01:03:47,870
All the way back, Zn now
is Xn plus Zn minus 1.

956
01:03:47,870 --> 01:03:50,990
OK, Zn is the sum of all
these random variables.

957
01:03:50,990 --> 01:03:54,030
So you get up n minus 1 of them,
and then you add the

958
01:03:54,030 --> 01:03:55,290
last one in.

959
01:03:55,290 --> 01:04:07,250
So it's Zn minus 1 plus Xn, so
its expected value of Xn plus

960
01:04:07,250 --> 01:04:12,700
Zn minus 1, given all the
stuff before that.

961
01:04:12,700 --> 01:04:17,160
The expected value of Xn, given
all this stuff, is what?

962
01:04:17,160 --> 01:04:20,640
Xn is independent of all the
other X's, therefore it's

963
01:04:20,640 --> 01:04:23,760
independent of all
the earlier Z's.

964
01:04:23,760 --> 01:04:27,840
And therefore, that's just
expected value of Xn.

965
01:04:27,840 --> 01:04:32,740
So we have the expected value of
Zn minus 1, given Zn minus

966
01:04:32,740 --> 01:04:34,380
1 back to Z1.

967
01:04:34,380 --> 01:04:39,130
What's expected value of Zn
minus 1, given Zn minus 1?

968
01:04:39,130 --> 01:04:42,596
Well, it's Zn minus 1.

969
01:04:42,596 --> 01:04:44,380
That's no problem there.

970
01:04:44,380 --> 01:04:47,200
So this is 0.

971
01:04:47,200 --> 01:04:53,340
So this is equal to Zn minus
1, as it's supposed to be.

972
01:04:53,340 --> 01:04:56,770
All of these things you ought
to go back and think them

973
01:04:56,770 --> 01:04:57,620
through yourself.

974
01:04:57,620 --> 01:05:01,120
Because the first time you look
at martingales, all of

975
01:05:01,120 --> 01:05:05,040
this stuff, it's all pretty
easy, but it all looks a

976
01:05:05,040 --> 01:05:07,860
little strange at first.

977
01:05:07,860 --> 01:05:10,260
The next one is sums
of arbitrary

978
01:05:10,260 --> 01:05:12,540
dependent random variables.

979
01:05:12,540 --> 01:05:15,270
They're not quite arbitrary.

980
01:05:15,270 --> 01:05:19,900
Suppose you have a sequence of
random variables, X sub i, i

981
01:05:19,900 --> 01:05:21,910
greater or equal to 1.

982
01:05:21,910 --> 01:05:25,970
And they satisfy the expected
value of Xi, given all the

983
01:05:25,970 --> 01:05:30,310
earlier X of i's
is equal to 0.

984
01:05:30,310 --> 01:05:33,060
It's similar to what
a martingale is.

985
01:05:33,060 --> 01:05:37,350
But here, we're just saying
the Xi's all have expected

986
01:05:37,350 --> 01:05:39,530
value of 0.

987
01:05:39,530 --> 01:05:45,350
And Zn, the sum of these,
has to be a martingale.

988
01:05:45,350 --> 01:05:48,380
And I'm not going to write
all a proof of that.

989
01:05:48,380 --> 01:05:55,020
I mean, this proof is really
the same as this proof.

990
01:05:55,020 --> 01:05:59,110
This is really a pretty
important thing, because given

991
01:05:59,110 --> 01:06:06,180
any martingale, you can always
look at the partial sums

992
01:06:06,180 --> 01:06:09,370
between the terms of
the martingale--

993
01:06:09,370 --> 01:06:13,120
namely, given Z1, Z2,
up to Z sub n.

994
01:06:13,120 --> 01:06:16,440
You can always look
at Z2 minus Z1.

995
01:06:16,440 --> 01:06:18,970
You can look at Z3 minus Z2.

996
01:06:18,970 --> 01:06:23,980
You could at Z4 minus
Z3, and so forth.

997
01:06:23,980 --> 01:06:26,910
And each of the Z's is
just the sum of those

998
01:06:26,910 --> 01:06:28,380
other random variables.

999
01:06:28,380 --> 01:06:32,810
So given any martingale in the
world, you can always define

1000
01:06:32,810 --> 01:06:37,920
the set of arbitrary depending
random variables would satisfy

1001
01:06:37,920 --> 01:06:39,970
this rule here.

1002
01:06:39,970 --> 01:06:45,900
So Zn, in this case,
is a martingale.

1003
01:06:45,900 --> 01:06:50,690
And if Zn is a martingale, you
can always define a set of

1004
01:06:50,690 --> 01:06:55,450
random variables, which
satisfy this property.

1005
01:06:55,450 --> 01:06:58,920
I think it's almost easier to
see what a random variable

1006
01:06:58,920 --> 01:07:01,430
really has to do with gambling,
which is where it

1007
01:07:01,430 --> 01:07:05,420
started, by looking at this.

1008
01:07:05,420 --> 01:07:08,790
This is not your capital
at time n.

1009
01:07:08,790 --> 01:07:13,750
This is how much you win
or lose at time i.

1010
01:07:13,750 --> 01:07:20,270
And what it's saying is your
winnings or losings at time i

1011
01:07:20,270 --> 01:07:25,560
has zero mean independent of
everything in the past.

1012
01:07:25,560 --> 01:07:28,770
In other words, in a fair game,
you can bet whatever you

1013
01:07:28,770 --> 01:07:34,470
want to and depending on what
you bet, that's the expected

1014
01:07:34,470 --> 01:07:37,940
amount you get on that trial.

1015
01:07:37,940 --> 01:07:40,590
And that's what this says.

1016
01:07:40,590 --> 01:07:43,860
This says essentially, you're
applying a fair game.

1017
01:07:43,860 --> 01:07:47,100
So martingales really have
to do is fair games.

1018
01:07:47,100 --> 01:07:50,390
If you can find fair games,
why, that's great.

1019
01:07:50,390 --> 01:07:54,200
But we always look for games
where we have an edge.

1020
01:07:54,200 --> 01:07:56,585
But what you want to avoid
is games where Las

1021
01:07:56,585 --> 01:07:58,630
Vegas has an edge.

1022
01:07:58,630 --> 01:08:01,800
OK so, that's a general one.

1023
01:08:04,320 --> 01:08:10,790
Here's an interesting one,
because I think this is an

1024
01:08:10,790 --> 01:08:13,470
example which you can use.

1025
01:08:13,470 --> 01:08:15,820
I mean, in any field you
study, there are always

1026
01:08:15,820 --> 01:08:20,760
generic examples, which can be
used to generate counter

1027
01:08:20,760 --> 01:08:24,899
examples to any simple thing
you might want to think of.

1028
01:08:24,899 --> 01:08:27,850
And this is to me the most
interesting one of those for

1029
01:08:27,850 --> 01:08:29,880
martingales.

1030
01:08:29,880 --> 01:08:34,800
Suppose that Xi is the product
of two random variables--

1031
01:08:34,800 --> 01:08:40,890
one is either plus 1 or minus 1,
each with probability 1/2.

1032
01:08:40,890 --> 01:08:44,510
And the other one, Y sub i is
anything it wants to be.

1033
01:08:44,510 --> 01:08:46,609
I don't care what Y sub i is.

1034
01:08:46,609 --> 01:08:50,810
Y sub i is non-negative, might
as well make it non-negative.

1035
01:08:50,810 --> 01:08:51,540
I don't care about it.

1036
01:08:51,540 --> 01:08:54,670
I don't care how it's related
to all the other Y sub i's.

1037
01:08:54,670 --> 01:08:58,200
All I want is the that the U sub
i's are all independent of

1038
01:08:58,200 --> 01:09:00,999
all the Y sub i's.

1039
01:09:00,999 --> 01:09:03,590
And what happens then?

1040
01:09:03,590 --> 01:09:07,420
I take the expected value of X
sub i, give it anything in the

1041
01:09:07,420 --> 01:09:09,529
past, and what do I get?

1042
01:09:12,100 --> 01:09:15,450
U sub i is independent
of Y sub i.

1043
01:09:15,450 --> 01:09:19,229
And therefore, the expected
value of U sub i times Y sub i

1044
01:09:19,229 --> 01:09:24,330
is expected value of U sub
i, which is what--

1045
01:09:24,330 --> 01:09:27,665
plus 1 or minus 1 of probability
1/2 each.

1046
01:09:27,665 --> 01:09:31,880
The expected value of U
sub i is equal to 0.

1047
01:09:31,880 --> 01:09:35,050
That makes expected value
of X sub i of 0

1048
01:09:35,050 --> 01:09:39,080
whatever the past is.

1049
01:09:39,080 --> 01:09:41,716
So you automatically
have the--

1050
01:09:46,676 --> 01:09:49,050
I don't know what to call in
them, the terms between the

1051
01:09:49,050 --> 01:09:51,640
terms of a martingale--

1052
01:09:51,640 --> 01:09:55,990
the interarrival terms,
so to speak.

1053
01:09:55,990 --> 01:09:59,500
I mean, it's like those
for a renewal process.

1054
01:09:59,500 --> 01:10:02,130
Those terms always
have mean 0.

1055
01:10:02,130 --> 01:10:08,180
And therefore, the sums of
these turn out to be this

1056
01:10:08,180 --> 01:10:11,000
simple kind of martingale.

1057
01:10:11,000 --> 01:10:14,910
So that's a nice martingale to
use as counter examples for

1058
01:10:14,910 --> 01:10:17,350
almost anything.

1059
01:10:17,350 --> 01:10:19,730
The next one is product
for martingales.

1060
01:10:19,730 --> 01:10:24,970
Product for martingales are
things we use quite a bit too,

1061
01:10:24,970 --> 01:10:28,240
because now when we're using
generating functions, we're in

1062
01:10:28,240 --> 01:10:31,200
the habit of multiplying
things together.

1063
01:10:31,200 --> 01:10:34,320
And that's a useful
thing to do.

1064
01:10:34,320 --> 01:10:41,000
So the expected value of Z to
the n, given Z to the n minus

1065
01:10:41,000 --> 01:10:48,590
1, now to Z1, where Zn is
this product of terms.

1066
01:10:48,590 --> 01:10:55,410
OK, Z sub n then is equal to
Xn times Z sub n minus 1,

1067
01:10:55,410 --> 01:10:56,900
which is what we're
doing here.

1068
01:10:56,900 --> 01:11:01,180
Expected value of Zn conditional
on the past is the

1069
01:11:01,180 --> 01:11:06,600
expected value of Xn times Zn
minus 1, conditional on the

1070
01:11:06,600 --> 01:11:13,170
past, Xn and Zn minus 1.

1071
01:11:19,750 --> 01:11:25,790
Oh, the expected value of Xn
for any given value of Zn

1072
01:11:25,790 --> 01:11:29,770
minus 1, all the way back,
is just the expected

1073
01:11:29,770 --> 01:11:32,750
value of X sub n.

1074
01:11:32,750 --> 01:11:36,510
So we have expected value of X
sub n times the expected value

1075
01:11:36,510 --> 01:11:40,970
of Z sub n minus 1, given
Zn n minus 1 down to Z1.

1076
01:11:40,970 --> 01:11:43,200
So that's just Zn minus 1.

1077
01:11:46,170 --> 01:11:51,260
Ah, the missing quantity,
fortunately I wrote it here.

1078
01:11:51,260 --> 01:11:53,675
The X sub i's are unit means
random variables.

1079
01:11:57,990 --> 01:11:58,880
And they're IIDs.

1080
01:11:58,880 --> 01:12:00,700
They're independent
of each other.

1081
01:12:00,700 --> 01:12:05,260
And since the X sub i's are
independent of each other, X

1082
01:12:05,260 --> 01:12:09,140
sub n is independent
of Xn minus 1, all

1083
01:12:09,140 --> 01:12:12,110
the way back to X1.

1084
01:12:12,110 --> 01:12:16,080
Zn minus 1 back to Z1
is a function of Xn

1085
01:12:16,080 --> 01:12:19,140
minus 1, down to X1.

1086
01:12:19,140 --> 01:12:24,090
So Xn is independent of all
those previous Z's also.

1087
01:12:24,090 --> 01:12:27,560
That's why I could split
this apart in this way.

1088
01:12:27,560 --> 01:12:31,350
And suddenly I wind up with
Zn n minus 1 again.

1089
01:12:31,350 --> 01:12:35,390
So product form martingales
work.

1090
01:12:35,390 --> 01:12:39,010
Special form of product
form martingales--

1091
01:12:39,010 --> 01:12:44,930
this again is favored counter
example for when you can and

1092
01:12:44,930 --> 01:12:48,530
can't get around with
going to limits and

1093
01:12:48,530 --> 01:12:51,040
interchanging limits.

1094
01:12:51,040 --> 01:12:53,360
And it's a simple one.

1095
01:12:53,360 --> 01:12:56,740
Suppose that X sub i's
are IID, as in

1096
01:12:56,740 --> 01:12:59,790
the previous example.

1097
01:12:59,790 --> 01:13:02,735
And they're [INAUDIBLE]
probably 2 or 0.

1098
01:13:09,010 --> 01:13:12,820
I mean, this is a game
you often play--

1099
01:13:12,820 --> 01:13:14,190
double or nothing.

1100
01:13:14,190 --> 01:13:15,950
You start out with dollar.

1101
01:13:15,950 --> 01:13:16,890
You play the game.

1102
01:13:16,890 --> 01:13:18,780
If you win, you get $2.

1103
01:13:18,780 --> 01:13:21,300
If you lose, you're broke.

1104
01:13:21,300 --> 01:13:24,210
If you win, you play your $2.

1105
01:13:24,210 --> 01:13:26,020
If you win again, you have $4.

1106
01:13:26,020 --> 01:13:27,760
If you lose, you're broke.

1107
01:13:27,760 --> 01:13:28,950
You play again.

1108
01:13:28,950 --> 01:13:30,460
If you win, you have $8.

1109
01:13:30,460 --> 01:13:33,020
If you lose again,
you're broke.

1110
01:13:33,020 --> 01:13:39,140
So the probability that Z sub n,
which is your capital after

1111
01:13:39,140 --> 01:13:43,640
n trials, is equals to 2 to the
n, namely you've won all n

1112
01:13:43,640 --> 01:13:48,540
times, is 2 to the minus n.

1113
01:13:48,540 --> 01:13:52,840
And every other instance
you've lost.

1114
01:13:52,840 --> 01:13:57,370
So the probability that Zn is
equal to 0 is 1 minus 2

1115
01:13:57,370 --> 01:13:59,170
to the minus n.

1116
01:13:59,170 --> 01:14:02,660
So for each n, if you calculate
the expected value

1117
01:14:02,660 --> 01:14:07,420
of Z sub n, it's equal to 1.

1118
01:14:07,420 --> 01:14:12,970
Namely, with probability 2 to
the minus n, your capital is 2

1119
01:14:12,970 --> 01:14:15,850
to the n, so that's 1.

1120
01:14:15,850 --> 01:14:18,630
With all the other probability,
you have nothing.

1121
01:14:18,630 --> 01:14:24,070
So your expected value of Z sub
n is always equal to 1.

1122
01:14:24,070 --> 01:14:27,420
That's what this product
form martingale says.

1123
01:14:27,420 --> 01:14:29,820
And this is a product
form martingale.

1124
01:14:29,820 --> 01:14:36,150
However, the limit as n goes to
infinity of Zn is equal to

1125
01:14:36,150 --> 01:14:39,130
0 with probability 1.

1126
01:14:39,130 --> 01:14:43,300
If you play double or nothing,
eventually you lose.

1127
01:14:43,300 --> 01:14:45,920
And then you're wiped out.

1128
01:14:45,920 --> 01:14:47,830
In other words, there's no real
purpose to playing the

1129
01:14:47,830 --> 01:14:50,720
game, because eventually
you lose.

1130
01:14:50,720 --> 01:14:53,620
If you're playing with somebody
else, and they're

1131
01:14:53,620 --> 01:14:56,500
playing double or nothing, then
of course you get their

1132
01:14:56,500 --> 01:14:58,530
money eventually.

1133
01:14:58,530 --> 01:15:02,890
Or you go broke and the bank
that you bank at fails, and

1134
01:15:02,890 --> 01:15:05,240
all that stuff.

1135
01:15:05,240 --> 01:15:06,810
We won't worry about that.

1136
01:15:06,810 --> 01:15:10,220
OK, but the point of this is
that the limit as n goes to

1137
01:15:10,220 --> 01:15:13,690
infinity of Zn is equal to 0.

1138
01:15:13,690 --> 01:15:17,020
And the limit of the expected
value of Z sub

1139
01:15:17,020 --> 01:15:19,920
n is equal to 1.

1140
01:15:19,920 --> 01:15:24,870
And therefore, the limit of Zn
and the expected value of the

1141
01:15:24,870 --> 01:15:26,130
limit of Zn.

1142
01:15:26,130 --> 01:15:29,260
The expected value of the
limit of Zn is 0.

1143
01:15:29,260 --> 01:15:33,550
The limit as expected value
of Zn is equal to one.

1144
01:15:33,550 --> 01:15:36,530
So this is a case where you
can't interchange limit and

1145
01:15:36,530 --> 01:15:37,920
expectation.

1146
01:15:37,920 --> 01:15:41,690
It's an easy one to keep in
mind, because we all know

1147
01:15:41,690 --> 01:15:43,740
about playing double
or nothing.

1148
01:15:50,630 --> 01:15:53,500
Might as well define
submartingales and

1149
01:15:53,500 --> 01:15:55,540
supermartingales.

1150
01:15:55,540 --> 01:16:00,800
Because the first thing to know
about them is they're

1151
01:16:00,800 --> 01:16:03,780
like martingales, except
they're defined with

1152
01:16:03,780 --> 01:16:06,220
inequalities.

1153
01:16:06,220 --> 01:16:10,580
And for a submartingale, the
expected value of Zn, given

1154
01:16:10,580 --> 01:16:12,970
all the previous terms,
is greater than or

1155
01:16:12,970 --> 01:16:15,560
equal to Zn minus 1.

1156
01:16:15,560 --> 01:16:18,520
So submartingales go up.

1157
01:16:18,520 --> 01:16:20,630
Supermartingales are
the opposite.

1158
01:16:20,630 --> 01:16:23,350
Supermartingales goes down.

1159
01:16:23,350 --> 01:16:27,930
What else could you expect from
a mathematical theory?

1160
01:16:27,930 --> 01:16:29,410
Things that should
go up, go down.

1161
01:16:29,410 --> 01:16:31,500
Things that should
go down, go up.

1162
01:16:31,500 --> 01:16:34,110
Only thing you have to remember
about submartingales

1163
01:16:34,110 --> 01:16:39,490
and supermartingales is you
figure out what terminology

1164
01:16:39,490 --> 01:16:43,350
should have been used, and you
remember the terminology they

1165
01:16:43,350 --> 01:16:45,750
use was the opposite of
what they should use.

1166
01:16:51,520 --> 01:16:52,980
I don't know whether I've
ever seen stupider

1167
01:16:52,980 --> 01:16:54,720
terminology than this.

1168
01:16:54,720 --> 01:16:57,890
And someone once explained the
reasoning for it, and the

1169
01:16:57,890 --> 01:16:59,140
reasoning was stupid too.

1170
01:17:01,990 --> 01:17:03,786
So there's no excuse
for that one.

1171
01:17:08,130 --> 01:17:11,680
We're only going to refer to
submartingales in what we're

1172
01:17:11,680 --> 01:17:15,330
doing, partly because that's
where most of the

1173
01:17:15,330 --> 01:17:16,980
neat results are.

1174
01:17:16,980 --> 01:17:24,920
And the other thing is if
you have to deal with a

1175
01:17:24,920 --> 01:17:28,350
supermartingale, what you might
as well do is instead of

1176
01:17:28,350 --> 01:17:29,430
dealing with a sequence--

1177
01:17:29,430 --> 01:17:30,850
Z1, Z2--

1178
01:17:30,850 --> 01:17:35,130
deal with the sequence minus
Z1, minus Z2, and so forth.

1179
01:17:35,130 --> 01:17:39,290
And if you change the sign on
all the terms, then you change

1180
01:17:39,290 --> 01:17:43,840
supermartingales into
submartingales and vice versa.

1181
01:17:43,840 --> 01:17:47,510
You don't really have to
deal with both of them.

1182
01:17:47,510 --> 01:17:57,090
Let me talk briefly about an
inequality that I'm sure most

1183
01:17:57,090 --> 01:17:58,450
of you heard of.

1184
01:17:58,450 --> 01:18:00,480
How many people have heard
of Jensen's Inequality?

1185
01:18:03,680 --> 01:18:06,930
Maybe half of you,
so not everyone.

1186
01:18:06,930 --> 01:18:10,700
Well, it's one of the
main work horses

1187
01:18:10,700 --> 01:18:12,540
of probability theory.

1188
01:18:16,730 --> 01:18:21,160
Even though we haven't seen it
yet this term, you will see it

1189
01:18:21,160 --> 01:18:22,990
many times.

1190
01:18:22,990 --> 01:18:26,680
So what a convex function is.

1191
01:18:26,680 --> 01:18:31,210
A convex function in simple
minded terms is something, a

1192
01:18:31,210 --> 01:18:33,790
convex function from r into r.

1193
01:18:33,790 --> 01:18:40,700
A real value convex function
is a function which has a

1194
01:18:40,700 --> 01:18:43,250
positive second derivative
everywhere.

1195
01:18:43,250 --> 01:18:46,760
So it curves down and
comes back up again.

1196
01:18:46,760 --> 01:18:49,730
Since you also want to talk
about functions which don't

1197
01:18:49,730 --> 01:18:53,740
have second derivatives,
you want something more

1198
01:18:53,740 --> 01:18:55,110
general than that.

1199
01:18:55,110 --> 01:18:58,760
So you go from derivatives.

1200
01:18:58,760 --> 01:19:01,260
You go back to your
high school ideas,

1201
01:19:01,260 --> 01:19:02,900
and you draw a picture.

1202
01:19:02,900 --> 01:19:07,060
And function is convex.

1203
01:19:07,060 --> 01:19:13,660
If all the tangents to the curve
lie not strictly below,

1204
01:19:13,660 --> 01:19:17,530
but all the tangents can curve
lie beneath the curve, so

1205
01:19:17,530 --> 01:19:20,800
wherever you draw a tangent,
you get something which

1206
01:19:20,800 --> 01:19:24,530
doesn't cross the curve.

1207
01:19:24,530 --> 01:19:29,550
Magnitude of X is a convex
function of X. Magnitude X as

1208
01:19:29,550 --> 01:19:32,120
a function of X looks
like this.

1209
01:19:32,120 --> 01:19:34,990
You go down or up.

1210
01:19:34,990 --> 01:19:39,740
And all tangents to this, this
goes off to infinity and this

1211
01:19:39,740 --> 01:19:40,790
goes off to infinity.

1212
01:19:40,790 --> 01:19:42,440
So there's no way to
get something like

1213
01:19:42,440 --> 01:19:44,670
that in this tangent.

1214
01:19:44,670 --> 01:19:46,540
So you have one tangent here.

1215
01:19:46,540 --> 01:19:48,870
You have a bunch of tangents
along here.

1216
01:19:48,870 --> 01:19:50,810
And you have one
tangent there.

1217
01:19:50,810 --> 01:19:52,840
And they all lie below
the curve.

1218
01:19:52,840 --> 01:19:56,320
So X bar is a convex
function too.

1219
01:19:56,320 --> 01:20:02,390
And Jensen's Inequality says if
H is convex, and if Z is a

1220
01:20:02,390 --> 01:20:07,970
random variable, it has finite
expectation, then H of the

1221
01:20:07,970 --> 01:20:12,470
expected value of Z is less than
or equal to the expected

1222
01:20:12,470 --> 01:20:16,960
value of H of Z. You can
interchange expected value and

1223
01:20:16,960 --> 01:20:21,020
function with inequality
like this, if

1224
01:20:21,020 --> 01:20:23,770
the function is convex.

1225
01:20:23,770 --> 01:20:25,180
Now, why is this true?

1226
01:20:29,430 --> 01:20:33,950
You can see why it's true
automatically, if you're

1227
01:20:33,950 --> 01:20:38,600
dealing with a random variable
that has only two values.

1228
01:20:38,600 --> 01:20:45,600
If you have two values for
the random variable--

1229
01:20:45,600 --> 01:20:47,680
Z is a random variable here.

1230
01:20:47,680 --> 01:20:51,630
You have one variable here,
one value here, one sample

1231
01:20:51,630 --> 01:20:53,660
value here.

1232
01:20:53,660 --> 01:21:00,130
Look at what the expected
value of H of Z is.

1233
01:21:00,130 --> 01:21:06,640
The expected value of H of Z is
the expected value of this

1234
01:21:06,640 --> 01:21:10,173
and this with the appropriate
probability put in on it, so

1235
01:21:10,173 --> 01:21:12,770
at some point that lies
on the straight line

1236
01:21:12,770 --> 01:21:15,990
between here and there.

1237
01:21:15,990 --> 01:21:21,910
When you look at the H of the
expect the value, then you

1238
01:21:21,910 --> 01:21:24,340
find the expected value
along here.

1239
01:21:24,340 --> 01:21:27,200
You can think of finding it
along the straight line here.

1240
01:21:27,200 --> 01:21:29,960
And then it's that
point there.

1241
01:21:29,960 --> 01:21:40,500
So since the curve is convex,
the H of the expected value of

1242
01:21:40,500 --> 01:21:43,870
Z is there's always
a bunch of points.

1243
01:21:43,870 --> 01:21:45,530
Average them, which
lie on a straight

1244
01:21:45,530 --> 01:21:47,820
line beneath the curve.

1245
01:21:47,820 --> 01:21:51,270
And the expected value of H
of Z is taking the average

1246
01:21:51,270 --> 01:21:53,220
directly along the curve.

1247
01:21:53,220 --> 01:21:56,280
So you get this boosting
up everywhere.

1248
01:21:56,280 --> 01:22:03,570
It's like saying that the
absolute value of expected

1249
01:22:03,570 --> 01:22:09,470
value of Z is less than or equal
to the expected value of

1250
01:22:09,470 --> 01:22:14,610
the absolute value of Z.

1251
01:22:14,610 --> 01:22:22,242
And I think I will stop there
instead of going on, because

1252
01:22:22,242 --> 01:22:24,410
we had a lot of new
things today.

1253
01:22:24,410 --> 01:22:30,390
And somehow sequential detection
always wears one's

1254
01:22:30,390 --> 01:22:34,590
mind out in a short
period of time.

1255
01:22:34,590 --> 01:22:35,840
That should be enough.