There are 2 levels of inference: model fitting and model comparison. In model fitting, assuming a model is true (say \(\mathcal{H}_i\)), fit the model to the data by inferring what values its free parameters should possibly take.

\begin{equation} P\left(\mathbf{w} | D, \mathcal{H}_{i}\right)=\frac{P\left(D | \mathbf{w}, \mathcal{H}_{i}\right) P\left(\mathbf{w} | \mathcal{H}_{i}\right)}{P\left(D | \mathcal{H}_{i}\right)} \end{equation}

The normalizing constant is irrelevant to the first level of inference. It is common to use gradient-based methods to find the maximum of the posterior \(\mathbf{w}_{\text{MP}}\). Error bars for these parameters can be obtained by evaluating the Hessian at \(\mathbf{w}_{\text{MP}}\), \(\mathbf{A}=-\nabla \nabla \ln P\left(\mathbf{w} | D, \mathcal{H}_{i}\right)\), and Taylor-expanding the log posterior probability with \(\Delta \mathbf{w}=\mathbf{w}-\mathbf{w}_{\mathrm{MP}}\):

\begin{equation} P\left(\mathbf{w} | D, \mathcal{H}_{i}\right) \simeq P\left(\mathbf{w}_{\mathrm{MP}} | D, \mathcal{H}_{i}\right) \exp \left(-1 / 2 \Delta \mathbf{w}^{\mathrm{T}} \mathbf{A} \Delta \mathbf{w}\right) \end{equation}

locally approximating the posterior as a Gaussian with covariance matrix \(\mathbf{A}^{-1}\).

In model comparison, we compare models in light of the data, assign some sort of preference.

Bayesian methods can consistently and quantitatively solve both types
of inferences, although adopting the Bayesian method for the first
type leads to similar results from orthodox statistical methods.
Orthodox statistical methods will find it difficult to perform model
comparisons, because *it is not possible simply to choose the model
that fits the data itself*. For example, maximum likelihood can fail
by choosing implausible, over-parameterized models that overfit the
data.

How do Bayesian methods perform model comparison? The posterior probability for each model is:

\begin{equation} P\left(\mathcal{H}_{i} | D\right) \propto P\left(D | \mathcal{H}_{i}\right) P\left(\mathcal{H}_{i}\right) \end{equation}

Hence, if we assign equal priors to the alternative models, models
\(\mathcal{H}_i\) are *ranked by evaluating the evidence*. If the
posterior is well approximated by a Gaussian, Bayesian model
comparison is a simple extension of maximum likelihood model
selection: the evidence is obtained by multiplying the best-fit
likelihood by the Occam factor, obtained from the determinant of the
covariance matrix \(\mathbf{A}^{-1}\) (the inverse Hessian).