STAT 310 DISCUSSION 9
TA: Yi Chai
Office: 1335N MSC
Email: [email protected]
Webpage: http://www.stat.wisc.edu/∼chaiyi
Office Hours: 11:00-12:00pm T and 1:00-2:00pm Th or by appointment
1. Bayesian analysis
•The Bayesian model for inference contains the statistical model fθ:θ∈Ω for the data s∈Sand
the prior probability distribution Π for θ.
•The prior describes the statistician’s beliefs about the true value of the parameter θbefore ob-
serving the data.
•Then one can compute the conditional density which is called the posterior distribution of θ
π(θ|s) = π(θ)f(s|θ)
m(s)
where m(s) = ZΩ
π(θ)fθ(s)dθ.
2. Dirichlet distribution
•Notation: θ= (θ1,· · · , θk−1)∼Dirichlet(α1,··· , αk);
•Probability density function:
p(θ) = Γ(α1+· · · +αk)
Γ(α1)· · · Γ(αk)θα1−1
1· · · θαk−1
k
where, αi>0, θi>0, α0=Pk
i=1 αi,Pk
i=1 θi= 1.
•E(θi) = αi
α0
;V ar(θi) = αi(α0−αi)
α2
0(α0+ 1) ;Cov(θi, θj) = −αiαj
α2
0(α0+ 1)
•specially when k=2, it reduces to Beta(α1, α2) distribution.
3. Examples
•Example 1 (7.1.1) Suppose that S={1,2}, Ω = {1,2,3}, and the class of probability distributions
for the response sis given by the following table.
s= 1 s= 2
f1(s) 1/2 1/2
f2(s) 1/3 2/3
f3(s) 3/4 1/4
If we use the prior π(θ) given by the table
θ= 1 θ= 2 θ= 3
π(θ) 1/5 2/5 2/5
then determine the posterior distribution of θfor each possible sample of size 2.
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