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Unformatted text preview: s’ rule, ignoring the constant de
nominator, we can express the posterior as:
p(Z, θ, φ|w, α, β ) ∝ p(w|Z, φ, θ, α, β )p(Z, θ, φ|α, β ) (21) We will look at each of these pieces and show that they have a compact 22 analytical form. m ni
mm p(w|Z, φ, θ, α, β ) = i=1 j =1
mm = i=1 j =1
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, φ|α, β ) = p(Z, θ|α)p(φ|β )
by conditional independence. Again considering each term, by the deﬁnition
of conditional probability:
p(Z, θ|α) = p(Z |θ, α)p(θ|α),
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(θ|α) = m
m p(θi |α) = m
m i=1 and
p(φ|β ) = i=1 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