Proof: Posterior density is proportional to joint likelihood
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Posterior density is proportional to joint likelihood
Metadata: ID: P90 | shortcut: post-jl | author: JoramSoch | date: 2020-05-05, 04:46.
Theorem: In a full probability model $m$ describing measured data $y$ using model parameters $\theta$, the posterior density over the model parameters is proportional to the joint likelihood:
\[\label{eq:post} p(\theta|y,m) \propto p(y,\theta|m) \; .\]Proof: In a full probability model, the posterior distribution can be expressed using Bayes’ theorem:
\[\label{eq:post-s1} p(\theta|y,m) = \frac{p(y|\theta,m) \, p(\theta|m)}{p(y|m)} \; .\]Applying the law of conditional probability to the numerator, we have:
\[\label{eq:post-s2} p(\theta|y,m) = \frac{p(y,\theta|m)}{p(y|m)} \; .\]Because the denominator does not depend on $\theta$, it is constant in $\theta$ and thus acts a proportionality factor between the posterior distribution and the joint likelihood:
\[\label{eq:post-qed} p(\theta|y,m) \propto p(y,\theta|m) \; .\]∎
Sources: Metadata: ID: P90 | shortcut: post-jl | author: JoramSoch | date: 2020-05-05, 04:46.