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The most common situation I'm familiar with is fitting any kind of Bayesian (regression) model to data. It's usually very easy to write down the likelihood (=sampling distribution of the data for given parameter values), as well as some prior distributions for the model parameters. The posterior distribution is proportional to their product, but it is ...

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The reason it is such a useful method reduces to to Bayes' formula: $$p(x \vert \text{data}) = \frac{p(\text{data} \vert x) p(x)}{p(\text{data})}$$ Typically, we denote $p(\text{data} \vert x)$ the likelihood function, $p(x)$ the prior distribution and in your notation $C = p(\text{data})$ is the marginal likelihood of the data. As statisticians or machine ...

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First, let me summarize the principle. Let $\boldsymbol{y}$ be the data (sample) and $p(\boldsymbol{y}|\boldsymbol{\theta})$ be the distribution of $\boldsymbol{y}$, parameterized by $\boldsymbol{\theta}$. Let take prior $p(\boldsymbol{\theta})$ for $\boldsymbol{\theta}$. After observing the sample (with the joint distribution \$p(\boldsymbol{y}|\boldsymbol{\...

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