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when choosing the prior distribution. The key notion is that if we choose
the ‘right’ prior for a particular likelihood function, then we can compute
the posterior without worrying about the integrating. We will formalize the
notion of conjugate priors and then see why they are useful.
Deﬁnition 3 (Conjugate Prior). Let F be a family of likelihood functions
and P a family of prior distributions. P is a conjugate prior to F if for
any likelihood function f ∈ F and for any prior distribution p ∈ P , the
corresponding posterior distribution p∗ satisﬁes p∗ ∈ P .
It is easy to ﬁnd the posterior when using conjugate priors because we know
it must belong to the same family of distributions as the prior.
Coin Flip Example Part 4. In our previous part of coin ﬂip example, we
were very wise to use a Beta prior for θ because the Beta distribution is the
conjugate prior to the Bernoulli distribution. Let us see what happens when
we compute the posterior using (2) for the likelihood and (7) for the prior:
θmH (1 − θ)m−mH θα−1 (1 − θ)β −1
p(θ|y ) = 1
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- Spring '12