Basic Bayesian Analysis

The classic formula (\(\theta\) = parameters):

\[p(\theta|\mathcal{Data}) = \frac{p(\mathcal{Data}|\theta)p(\theta)}{p(\mathcal{Data})}\]

Conceptually, we can think about it in different ways

\[\mathrm{posterior} \propto likelihood * prior\]

Standard methods you are already familiar with begin and end with likelihood

They can be seen as Bayesian analysis with uninformative priors

Alternatively, we can think also in terms of the posterior as the combination hypothesis, and evidence, in the form of data.

\[p(hypothesis|data) \propto p(data|hypothesis)p(hypothesis)\]

\[\text{updated belief} = \text{current evidence} * \text{prior belief or evidence}\]
Some key distinctions: