Proof: Joint likelihood is the product of likelihood function and prior density
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The Book of Statistical Proofs ▷
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Joint likelihood is product of likelihood and prior
Metadata: ID: P89 | shortcut: jl-lfnprior | author: JoramSoch | date: 2020-05-05, 04:21.
Theorem: Let there be a generative model $m$ describing measured data $y$ using model parameters $\theta$ and a prior distribution on $\theta$. Then, the joint likelihood is equal to the product of likelihood function and prior density:
\[\label{eq:jl} p(y,\theta|m) = p(y|\theta,m) \, p(\theta|m) \; .\]Proof: The joint likelihood is defined as the joint probability distribution of data $y$ and parameters $\theta$:
\[\label{eq:jl-def} p(y,\theta|m) \; .\]Applying the law of conditional probability, we have:
\[\label{eq:jl-qed} \begin{split} p(y|\theta,m) &= \frac{p(y,\theta|m)}{p(\theta|m)} \\ &\Leftrightarrow \\ p(y,\theta|m) &= p(y|\theta,m) \, p(\theta|m) \; . \end{split}\]∎
Sources: Metadata: ID: P89 | shortcut: jl-lfnprior | author: JoramSoch | date: 2020-05-05, 04:21.