Occam’s razor is the principle that states a preference for simpler models. This is not just a philosophical choice: Bayesian probabilistic inference automatically embodies this principle, quantitatively. To see this, we evaluate 2 alternative theories \(\mathcal{H}_1\) and \(\mathcal{H}_2\), in light of observing data \(\mathcal{D}\).

\begin{equation} \frac{P(\mathcal{H}_1|\mathcal{D})}{P(\mathcal{H}_2|\mathcal{D})} = \frac{P(\mathcal{H}_1)}{P(\mathcal{H}_2)} \frac{P(\mathcal{D}|\mathcal{H}_1)}{P(\mathcal{D}|\mathcal{H}_2)} \end{equation}

The ratio \(\frac{P(\mathcal{H}_1)}{P(\mathcal{H}_2)}\) denotes how much our initial beliefs favoured \(\mathcal{H}_1\) over \(\mathcal{H}_2\). The second ratio expresses how well relatively the observed data \(\mathcal{D}\) were predicted by the 2 hypotheses.

Simple models make precise computations, while complex models spread their predictive probabilities more thinly over their larger hypothesis space. In the case where the data are compatible with both theories, the simpler \(\mathcal{H}_1\) would turn out to be more probable than the more complex \(\mathcal{H}_2\). Hence the second term automatically embodies the Occam’s razor.

## Gelman on the Occam Factor

Source: David MacKay and Occam’s Razor « Statistical Modeling, Causal Inference, and …

Gelman is not fond of Mackay’s above argument about Bayesian inference embodying Occam razor. His argument seems to be about wanting to keep more complex models:

once I’ve set up a model I’d like to keep all of it, maybe shrinking some parts toward zero but not getting rid of coefficients entirely.

I still don’t see a contradiction with Mackay’s proposed argument. Maybe I’m missing something…