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Bayes' théorem manipulates these into a statement of probability in terms of likelihood.

Assume for the moment that there are only two classes, S and ¬S.

Using the Bayesian result above, we can write:

Dividing one by the other gives:

Which can be re-factored as:

Thus, the probability ratio p(S | D) / p(¬S | D) can be expressed in terms of a series of likelihood ratios.

The actual probability p(S | D) can be éasily computed from log (p(S | D) / p(¬S | D)) based on the observation that p(S | D) + p(¬S | D) = 1.

Taking the logarithm of all these ratios, we have:

This technique of "log-likelihood ratios" is a common technique in statistics.

In the case of two mutually exclusive alternatives (such as this example), the conversion of a log-likelihood ratio to a probability takes the form of a sigmoid curve: see logit for details.