Binomial Distribution & Bayes’ Theorem

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 more heads in 4 coin tosses?

• Give an concrete illustration of p(D|H) and p(H|D). Why might these be different?

Probability of Binary Events

• Probability of success = p

• p(success) = p

• Probability of failure = q

• p(failure) = q

• p+q = 1

• q = 1-p

• Probability – long run relative frequency

Permutations & Combinations 1• Suppose we flip a coin 2 times• H H• H T• T H• T T• Sample space shows 4 possible outcomes or

sequences. 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.

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 2

trials is ¼ = 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?]

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

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 N objects,

regardless of order. Say 2 heads from 5 trials.

)!(!

!

rNr

N

r

N

1025)123)(12(

12345

)!25(!2

!5

2

5

x

xxx

xxxx

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

NpNrxp rNr ,...2,1,),;(

Binomial 2Nrqp

r

NpNrxp rNr ,...2,1,),;(

5 .03125

4 .15625

3 .3125

2 .3125

1 .15625

0 .03125

55 5.5

5)5.,5;5(

ppNXp

qpqppNXp 414 54

5)5.,5;4(

2323 103

5)5.,5;3( qpqppNXp

3232 102

5)5.,5;2( qpqppNXp

4141 51

5)5.,5;1( qpqppNXp

550

0

5)5.,5;0( qqppNXp

Binomial 3

• Flip coins and compare observed to expected frequencies

Binomial 4

• Find expected frequencies for number of 1s from a 6-sided die in five rolls.

Binomial 5

• When p is .5, as N increases, the binomial approximates the Normal.

1086420Number Heads

300

200

100

0

Rel

ativ

e F

requ

ency

(N

umbe

r of

Com

bos)

Binomial N = 10 p = .5

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 more heads in 4 coin tosses?

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 probability – personal belief or data. Input.Likelihood – likelihood of data given hypothesis.Posterior probability – probability of hypothesis given data.

Scientists are interested in substantive hypotheses, e.g., does Nicorette help people stop smoking. The p level that comes from the study is the probability of the sample data given the hypothesis, not the probability of the hypothesis given the data. That is

)|()|( hypothesisdatapdatahypothesisp

Bayes Theorem (2)Bayes theorem is old and mathematically correct. But its use 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)

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 HpHypHpHyp

HpHypyHp

704.)25)(.111(.)75)(.088(.

)75)(.088(.)|6(.

yp

Bayes Theorem (4)

Problems with choice of prior. Handled by empirical 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 the probability 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.

Review

Give an concrete illustration of p(D|H) and p(H|D). Why might these be different?