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Unformatted text preview: s’ rule, ignoring the constant de
nominator, we can express the posterior as:
p(Z, θ, φw, α, β ) ∝ p(wZ, φ, θ, α, β )p(Z, θ, φα, β ) (21) We will look at each of these pieces and show that they have a compact 22 analytical form. m ni
mm p(wZ, φ, θ, α, β ) = i=1 j =1
m ni
mm = i=1 j =1
m ni
mm = p(wi,j Z, φ, θ, α, β ) By iid assumption
p(wi,j zi,j , φ) By conditional independence
Multinomial(wi,j ; φzi,j ) By deﬁnition. i=1 j =1 Also,
p(Z, θ, φα, β ) = p(Z, θα)p(φβ )
by conditional independence. Again considering each term, by the deﬁnition
of conditional probability:
p(Z, θα) = p(Z θ, α)p(θα),
where
m ni
mm p(Z θ, α) = p(Z θ) = p(zi,j θi ) = i=i j =1 m ni
mm Multinomial(zi,j ; θi ), i=i j =1 by conditional independence, iid, and deﬁnition as before. Further,
p(θα) = m
m p(θi α) = m
m i=1 and
p(φβ ) = i=1 K
m K
m Dirichlet(θi ; α), p(φk β ) = k =1 Dirichlet(φk ; β ). k =1 Plugging all of these pieces back into (21), we ob...
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This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.
 Spring '12
 CynthiaRudin

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