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1
Lecture #7
Induction Continued
AGENDA
I.
Probabilistic Induction
A. Reverend Thomas Bayes
B. Bayes Theorem and Induction
C. Probabilistic Induction Brain Teasers
I.
Logical Induction
A. Mathematical Induction
B. A Brain Teaser
I.
Human Inductive Heuristics
II.
Paradoxes of Induction
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I. Probabilistic Induction
Probability theory developed after plane geometry (Euclid, ca
300 B.C.); analytic geometry (Rene Descartes, 15961650);
calculus (Sir Isaac Newton, 16421727; Gottfried Leibniz)
Pascal (16231662), Fermat (16011665),
and Jacob
Bernoulli (16541705) were among the pioneers. Their results
came in part from assisting gamblers.
Fermat
16011665
Jacob Bernoulli
16541705
Pascal
16231662
3
Pascal and Fermat’s problem
•
Suppose a pair of dice is rolled until a boxcar occurs
(boxcar= double six). How many rolls are needed to bring
the probability over ½ that a boxcar will occur?
•
Well the probability that a boxcar will
occur in n tosses is
•
Well for n=25,
Pr(boxcar)=0.505,
about an equal bet.
n

=
36
35
1
rolls)
n
in
boxcar
one
least
at
Pr(
•
Hint: Probability at least one head in 3 flips=
1 prob (no heads in three flips)=1(1/2)
3
= 7/8
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A. Reverend Thomas Bayes
(17021761)
Bayes formulated a way to
inductively update the strength
of beliefs with evidence.
Beliefs are represented by probabilities and evidence changes
these probabilities. His formula involved conditional probability.
Belief (A given evidence E)=
(Likelihood of E)
=
Prior
Belief in A
Evidence E
Concerning A
Posterior Belief in
A given evidence
(Likelihood of E given belief A)(Prior belief A)
(Likelihood of
E)
(Likelihood of E given A)(prior belief A) +(likelihood E given not A)(prior belief not A)
5
Example of Bayesian Inference
Suppose there are two urns: U
I
has 2 red balls
and 1 blue ball, U
II
has 1 red and 3 blue.
A games master selects an urn at
random
and
takes a ball at
random
. The ball is
blue
. What
is your belief in the probability it came from U
I
?
Prior
: Pr(U
I
)=1/2.
Evidence
: Ball is
blue
Posterior
: Pr(U
I
blue)=?
I
II
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Simple Probability Facts From
Statistics
If A and B are nonzero probability events, i.e.
Then the conditional probability of B given A is
defined as
From which we have
Bayes theorem
:
0
)
Pr(
),
Pr(
B
A
)
Pr(
)
Pr(
)
Pr(
A
B
A
A
B
∩
=
)
Pr(
)
Pr(
)
Pr(
)
Pr(
)
Pr(
)
Pr(
A
B
B
A
A
B
A
A
B
⋅
=
∩
=
Prob both A and B occur
7
The Solution Using Bayes Theorem
The probability of the evidence is obtained by
simple probability theory:
(1/3)(1/2)+(3/4)(1/2)=13/24
13
/
4
)
24
/
13
(
)
2
/
1
(
)
3
/
1
(
(B)
Pr
)
U
Pr(
)
U
Pr(B
B)
U
Pr(
I
I
I
=
⋅
=
⋅
=
Posterior belief
probability
Probability
of evidence
given U
I
Probability
of evidence
Prior belief
probability
=
+
⋅
=
)
U
Pr(
)
U
B
Pr(
)
U
Pr(
)
U
Pr(B
B)
Pr(
II
II
I
I
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B. Bayes Theorem and Induction
Bayes thought of H as a scientific
hypothesis and E as the
evidence. It is often easy to calculate the probability of the
evidence given the hypothesis.
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This note was uploaded on 12/05/2010 for the course PSYCH Psy Beh F2 taught by Professor Williamh.batchelder during the Fall '10 term at UC Irvine.
 Fall '10
 WilliamH.BATCHELDER

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