MIT15_097S12_lec15

Let us see what happens when we compute the posterior

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Unformatted text preview: exibility 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 = (normalizat...
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