6 binomialnbayes
TRANSCRIPT
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Binomial Distribution &
Bayes Theorem
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Questions
What is a probability?
What is the probability of obtaining 2
heads in 4 coin tosses? What is the
probability of obtaining 2 or moreheads in 4 coin tosses?
Give an concrete illustration of p(D|H)
and p(H|D). Why might these bedifferent?
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Probability of Binary Events
Probability of success = p
p(success) = p
Probability of failure = q
p(failure) = q
p+q = 1
q = 1-p Probabilitylong run relative
frequency
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Permutations & Combinations
1 Suppose we flip a coin 2 times
H H
H T
T H T T
Sample space shows 4 possible outcomes orsequences. Each sequence is a permutation.
Order matters. There are 2 ways to get a total of one heads
(HT and TH). These are combinations.Order does NOT matter.
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Perm & Comb 2
HH, HT, TH, TT
Suppose our interest is Heads. If the coin is
fair, p(Heads) = .5; q = 1-p = .5.
The probability of any permutation for 2trials is = p*p, or p*q, or q*p, or q*q. All
permutations are equally probable.
The probability of exactly 1 head in any
order is 2/4 = .5 =
HT+TH/(HH+HT+TH+TT) [what is
probability of at least 1 head?]
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Perm & Comb 3
3 flips
HHH,
HHT, HTH, THH
HTT, THT, TTH
TTT
All permutations equally likely = p*p*p= .53 = .125 = 1/8.
p(1 Head) = 3/8
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Perm & Comb 4
Factorials: N!
4! = 4*3*2*1
3! = 3*2*1
Combinations: NCr
The number of ways of selecting r combinations of Nobjects, regardless of order. Say 2 heads from 5 trials.
)!(!!rNr
NrN
1025)123)(12(
12345
)!25(!2
!5
2
5
x
xxx
xxxx
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Binomial Distribution 1
Is a binomial distribution with
parameters N and p. N is the number of
trials, p is the probability of success.
Suppose we flip a fair coin 5 times; p =
q = .5
Nrqpr
N
pNrxprNr
,...2,1,),;(
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Binomial 2Nrqp
r
NpNrxp rNr ,...2,1,),;(
5 .03125
4 .15625
3 .3125
2 .3125
1 .15625
0 .03125
555.
5
5)5.,5;5(
ppNXp
qpqppNXp
414
54
5
)5.,5;4(
232310
3
5)5.,5;3( qpqppNXp
323210
2
5)5.,5;2( qpqppNXp
41415
1
5)5.,5;1( qpqppNXp
550
0
5)5.,5;0( qqppNXp
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Binomial 3
Flip coins and compare observed to
expected frequencies
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Binomial 4
Find expected frequencies for number
of 1s from a 6-sided die in five rolls.
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Binomial 5
When p is .5, as N increases, the
binomial approximates the Normal.
1086420Number Heads
300
200
100
0RelativeFrequency
(NumberofComb
os)
Binomial N = 10 p = .5
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Review
What is a probability?
What is the probability of obtaining 2
heads in 4 coin tosses? What is the
probability of obtaining 2 or moreheads in 4 coin tosses?
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Bayes Theorem (1)Bayesian statistics are about the revision of belief. Bayesian
statisticians look into statistically optimal ways of
combining new information with old beliefs.
Prior probabilitypersonal belief or data. Input.
Likelihoodlikelihood of data given hypothesis.
Posterior probabilityprobability of hypothesis given data.
Scientists are interested in substantive hypotheses, e.g.,
does Nicorette help people stop smoking. Thep level
that comes from the study is the probability of the sampledata given the hypothesis, not the probability of the
hypothesis given the data. That is
)|()|( hypothesisdatapdatahypothesisp
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Bayes Theorem (2)
Bayes theorem is old and mathematically correct. But itsuse is controversial. Suppose you have a hunch about the
null (H0) and the alternative (H1) that specifies the
probability of each before you do a study. The probabilities
p(H0
) and p(H1
) are priors. The likelihoods are p(y| H0
) and
p(y| H1). Standard p values. The posterior is given by:
)()|()()|(
)()|()|(
1100
000
HpHypHpHyp
HpHypyHp
p(H1|y)=1-p(H0|y)
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Bayes Theorem (3)
Suppose before a study is done that the two hypotheses are
H0: p =.80 and H1: p=.40 for the proportion of male grad
students. Before the study, we figure that the probability is
.75 that H0 is true and .25 That H1 is true. We grab 10 grad
students at random and find that 6 of 10 are male.Binomial applies.
111.)4.|6(.)|6(.
088.)8.|6(.)|6(.
25.)(;75.)(
1
0
10
ppHp
ppHp
HpHp
)()|()()|()()|()|(
1100
000
HpHypHpHypHpHypyHp
704.)25)(.111(.)75)(.088(.
)75)(.088(.)|6(.
yp
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Bayes Theorem (4)
Problems with choice of prior. Handled byempirical data or by flat priors. There are
Bayesian applications to more complicated
situations (e.g., means and correlations). Not
used much in psychology yet except in meta-analysis (empricial Bayes estimates) and
judgment studies (Taxis, etc). Rules for
exchangeability (admissible data) need to be
worked out.
Bayes theorem says we should revise our belief of theprobability that H0 is true from .75 to .70 based on new
data. Small change here, but can be quite large depending
on data and prior.
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Review
Give an concrete illustration of
p(D|H) and p(H|D). Why might
these be different?