102
with the same prognosis.
Appendix
Gibbs Sampling Scheme
Any set of imputed parameters θι,..., θn corresponds to a partition p = (Si,..., S⅛)
of the indices {1,..., n}. Let θ*,..., θ*κ the unique values in the sample in order of
appearance and define Sk = {i '■ θi = θ*k}.
Consider the Dirichlet process mixture model (Antoniak, 1974) in (3.9). Let
д~г — (flɪ,...,0i-ι,θi+1, ...,θn) denote all values but θi. Let K~l be the number of
unique values in θ~∖ let θ↑~∖ ..., θ*1^i be these different values in order of appearance
and p~l be the partition implied by θ~l. Use equation (4.5) with the n—1 observations
and let the observation i be the future observation (exchangeability allows us to
permutate the indices) to get:
κ~i
p[θi I θ~i] = fκ-41(p-i)g0(θi) + ∑ Λ(p-‰-O,
fc=l
where δx is a pointmass at x and go is the pdf corresponding to Go- Therefore,
p[θi I θ~i,yn] <xp[yn I θi,θ~i]p[θi I θ~i]
κ~i `
⅛.+ι(∏G0(¾) + ,
fc=l
κ~i
<xp[Vi I θl]fκ→+1{p~l')Go{θi') + V fk(p~l)p[yi I θi]δθ*-i(θi)
• ∙ K
fc=l
Using (4.5) we get:
κ~i / +1 V
p[θi I θ~∖ yn] oc ⅛p[yi I 0i]Go(0l) + £ #Sfc ) p[yi ∣ Λ-i(θi),
where I is the value of the categorical covariate (in our particular, the prognosis)
corresponding to the ith experimental unit (patient).