Bayesian Methods in Health Economics

54 Bayesian Methods in Health Economics

parameters (α, β) will give

p(θ | y) ∝L(θ) × p(θ)

∝ Beta(y +1,n− y +1)× Beta(α, β)

∝



θ

y

(1 − θ)

n−y



×



θ

α−1

(1 − θ)

β−1



= θ

y+α−1

(1 − θ)

n−y+β−1

,

which is the kernel of a Beta(y + α, n − y + β) distribution. Consequently,

the posterior distribution is still in the same functional form and the update

process from prior to posterior only involves a change in the hyper-parameters,

which are modified from (α, β) to (y + α, n −y + β) to incorporate the effect

of the observed data.

Table 2.3 shows some common conjugate models — cfr. Bernardo and

Smith (1999) for a taxonomic account of conjugated families. Throughout the

table, the ...