6 binomialnbayes

7/27/2019 6 BinomialnBayes

1/18

Binomial Distribution &

Bayes Theorem


2/18

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?


3/18

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


4/18

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.


5/18

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?]


6/18

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


7/18

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


8/18

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,),;(


9/18

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


10/18

Binomial 3

Flip coins and compare observed to

expected frequencies


11/18

Binomial 4

Find expected frequencies for number

of 1s from a 6-sided die in five rolls.


12/18

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


13/18

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?


14/18

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


15/18

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)


16/18

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


17/18

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.


18/18

Review

Give an concrete illustration of

p(D|H) and p(H|D). Why might

these be different?

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