Index: The Book of Statistical ProofsGeneral Theorems ▷ Bayesian statistics ▷ Probabilistic modeling ▷ Posterior density is proportional to joint likelihood

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:

\[\label{eq:post-qed} p(\theta|y,m) \propto p(y,\theta|m) \; .\]

Metadata: ID: P90 | shortcut: post-jl | author: JoramSoch | date: 2020-05-05, 04:46.