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Unformatted text preview: hyperparameters were set to a = 3, b = 1,
p(z |y ) ∼ Neg-Binomial 16,
7 6 4. Suppose y1 , . . . , yn is a random sample from a negative binomial distribution with parameters (r, θ),
where r is known and θ is unknown. Here, r denotes the number of successes and θ denotes the
Suppose θ ∼ Beta(α, β ).
(a) Derive the posterior distribution of θ.
The likelihood function of θ is given by
L ( θ ; y1 , . . . , y n ) = n
yi + r − 1
yi + r − 1 r
θ (1 − θ)yi =
i=1 θnr (1 − θ) n i=1 yi The posterior distribution of θ is given by
p(θ|y ) ∝ θnr (1 − θ) n i=1 yi α − 1 θ = θα+nr−1 (1 − θ)β + Thus, θ|y ∼ Beta(α + nr, β + n n (1 − θ)β −1 i=1 yi − 1 i=1 yi ). The prior and the posterior are in the same family of distribution, so the beta prior is conjugate
to the negative-binomial sampling model.
(b) Construct a 95% credible interval for θ using α = β = 1.
For the observed data, 10 i=1 yi = 75 and n...
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This note was uploaded on 02/22/2013 for the course MATH 604 taught by Professor Tadasee during the Spring '13 term at Georgetown.
- Spring '13