A common alternative to evaluate and compare models in the Bayesian world (at least in some of its countries) are Bayes factors. To understand what Bayes factors are, let's write Bayes' theorem one more time (we have not done so for a while!):

Here,  represents the data. We can make the dependency of the inference on a given   model explicit and write:

The term in the denominator is known as marginal likelihood (or evidence), as you may remember from the first chapter. When doing inference, we do not need to compute this normalizing constant, so in practice, we often compute the posterior up to a constant factor. However, for model comparison and model averaging, the marginal likelihood is an important quantity. If our main objective is to choose only one model, the best one, from a set of  models, we can just choose the one with the largest . As a general rule,  are tiny numbers and do not tell us too much on their own; like with information criteria, what matters are the relative values. So, in practice, people often compute the ratio of two marginal likelihoods, and this is called a Bayes factor:

When , model 0 explains data better than model 1.

Some authors have proposed tables with ranges to discretize and ease interpretation. For example the following bullet-list indicates the strength of the evidence, favoring model 0 against model 1:

• 1-3: Anecdotal
• 3-10: Moderate
• 10-30: Strong
• 30-100: Very strong
• > 100: Extreme

Remember, these rules are just conventions, simple guides at best. However, the results should always be put into context and should be accompanied with enough details that others could potentially check if they agree with our conclusions. The evidence that's necessary to make a claim is not the same in particle physics, or a court, or in a plan to evacuate a town to prevent hundreds of deaths.

Using  to compare models is totally fine if all of the models are assumed to have the same prior probability. Otherwise, we have to compute the posterior odds: