The name is absent

102

with the same prognosis.

Appendix

Gibbs Sampling Scheme

Any set of imputed parameters θι,..., θ_n 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,...,0_i-ι,θ_i+1, ...,θ_n) denote all values but θ_i. 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[θ_i I θ~ⁱ] = f_κ-₄₁(p-ⁱ)g₀(θ_i) + ∑ Λ(p-‰-O,
fc=l

where δ_x is a pointmass at x and go is the pdf corresponding to Go- Therefore,

p[θ_i I θ~ⁱ,yⁿ] <xp[yⁿ I θ_i,θ~ⁱ]p[θ_i I θ~ⁱ]

κ~ⁱ                  `

⅛.₊ι(∏G₀(¾) +             ,

fc=l
κ~ⁱ

<xp[Vi I θ_l]f_κ→₊₁{p~^l')Go{θ_i') + V fk(p~^l)p[y_i I θ_i]δ_θ*-i(θ_i)
• ∙                                              K

fc=l

Using (4.5) we get:

κ~ⁱ /    +1 V

p[θ_i I θ~∖ yⁿ] oc ⅛p[y_i I 0_i]G_o(0_l) + £ #S_fc           ) p[y_i ∣ Λ-_i(θ_i),

where I is the value of the categorical covariate (in our particular, the prognosis)
corresponding to the i^th experimental unit (patient).

<<   111   112   113   114   115   116   117   >>

More intriguing information