The Big Picture
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T H E B IG PIC T U R E
01
Bayes’s Theorem tells us how to update those credences when we
02
get some new information. Let’s say we get information in the form of
03
some new data, such as the number of cards our opponent draws. Then
04
for each jar, we remove a fraction of the sand corresponding to the likelihood 05
that we would not have obtained that data if the corresponding proposition
06
were correct. If we think our opponent would draw precisely one card only
07
10 percent of the time if they had a pair, we remove nine- tenths of the grains
08
of sand from the jar labeled “pair” when we see them draw a single card.
09
Then we do the analogous thing for all the other jars. At the end, our
10
grains-of-sand rule is once again true: the credence of proposition X is the
11
number of grains of sand in jar X divided by the total number in all the jars.
12
What this procedure does is to re-weight the prior credences by the like-
13
lihoods, in order to obtain posterior credences. We might start with a situ-
14
ation where several jars have approximately the same amount of sand,
15
corresponding to equal credences. But then we obtain some new informa-
16
tion, which would be likely under some propositions and unlikely under
17
some other ones. We remove just a little sand from the jars where the infor-
18
mation was likely, and a lot of sand from those where the information was
19
unlikely. We’re left with a relatively greater amount of sand in the more-
20
likely jars, corresponding to greater posterior credence for those proposi-
21
tions. Of course, if our prior credence in one proposition was incredibly
22
large compared to that for its competitors, we would have to remove a very
23
large amount of sand (collect data that was very unlikely under that propo-
24
sition) for that credence to become small. When priors are very large or very
25
small, the data has to be very surprising in order to shift our credences.
26
•
27
28
Consider a different scenario: you’re a high school student, you have a crush
29
on someone, and you want to ask them to the prom. The question is, will
30
they say yes, or no? So there are two different propositions: “Yes” (they will
31
go to the prom with you) and “No” (they won’t), and for each we have a prior
32
credence. Let’s be optimistic and assign credence 0.6 to Yes, and 0.4 to No.
33
(Clearly the total credences must always add up to 1.) We set up two jars of
34
sand, in which we place 60 grains in the Yes jar and 40 grains in the No jar.
35S
The total number of grains doesn’t matter, only the relative proportion.
36N
Our next step is to collect new information and update our priors by
76
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u PdA t I n g Ou R K nOW l E d g E
using likelihoods. You’re standing at your locker, and you see your crush
01
walking down the hall. Will they say hi, or just walk right by you? That
02
depends on how they think about you— they’re more likely to stop and say
03
hi if they’re also inclined to go with you to the prom than if they’re not so
04
inclined. Using your keen knowledge of human interaction, under proposi-
05
tion Yes they will stop and say hi 75 percent of the time, and walk right by
06
25 percent (maybe they were just distracted). But under proposition No, the
07
odds aren’t as good: 30 percent of the time they’ll say hi, and 70 percent
08
they’ll walk right by. Those are your likelihoods for various information to
09
be gathered under the different propositions. Time to collect some data and
10
update your credences!
11
Let’s say that your crush does, to your delight, stop and say hi. How does
12
that affect the chances that they would accept an invitation to the prom?
13
Reverend Bayes tells us to remove 25 percent of the sand from the Yes jar,
14
and 70 percent of the sand from the No jar (corresponding in each case to
15
16
17
Yes
No
18
19
20
Prior: 60 grains
40 grains
= 60% of total
= 40% of total
21
22
23
24
25
26
Update: Remove 25%
Remove 70%
27
= 15 grains
= 28 grains
28
29
30
31
32
Final: 45 grains
12 grains
33
= 79% of total
= 21% of total
34
S35
N36
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T H
T E
H B
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11
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14
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15
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16
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17
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