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Unformatted text preview: exp (φ(θ' )T ν ) dθ' ��
u(yi ) (dropping nonθ terms), i=1 which is in the same family as the prior (17), with the posterior hyperparam
eters being η + m and ν + m u(yi ).
i=1
Although this proof yields the form of the conjugate prior, we may not always
be able to compute the partition function. So we can’t always gain something
in practice from it.
It turns out that in general, the converse to Theorem 1 is true, and exponen
tial families are the only distributions with (nontrivial) conjugate priors.
Coin Flip Example Part 5. Returning again to the coin ﬂip example, let
us ﬁrst verify that the Bernoulli distribution is an exponential family:
p(yi θ) = θyi (1 − θ)1−yi
= exp (yi log θ + (1 − yi ) log(1 − θ))
= exp (yi log θ − yi log(1 − θ) + log(1 − θ))
θ
= (1 − θ) exp yi log
,
1−θ
and we see that the Bernoulli distribution is an exponential family according
θ
to (16) with f (yi ) = 1, g (θ) = 1 − θ, u(yi ) = yi , and φ(θ) = log 1−θ . Th...
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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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