MIT15_097S12_lec15

The appropriate likelihood function binomial gaussian

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Unformatted text preview: ularization can be interpreted as a Gaussian prior. Increasing C corresponds to increasing our certainty that θ should be close to zero. 3 Conjugate priors Although point estimates can be useful in many circumstances (and are used in many circumstances), our true goal in Bayesian analysis is often to find the full posterior, p(θ|y ). One reason for wanting the posterior is to be able to use the posterior predictive distribution of a yet unobserved data point: � � p(ym+1 |y ) = p(ym+1 |y, θ)p(θ|y )dθ = p(ym+1 |θ)p(θ|y )dθ, (13) 9 because ym+1 and y are conditionally independent given θ by assumption. We saw earlier that the posterior can be obtained in principle from the prior and the likelihood using Bayes’ rule, but that there is an integral in the denominator which often makes this intractable. One approach to circum­ venting the integral is to use conjugate priors. The appropriate likelihood function (Binomial, Gaussian, Poisson, Bernoulli,...) is typically clear from the data. However, there is a great deal of ...
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