Hypothesis test (discrete case)

Although in conventional statistics books this argument is usually dealt with in one of the later chapters, in the Bayesian approach it is so natural that it is in fact the first application, as we have seen in the above examples. We summarize here the procedure:

• probabilities are attributed to the different hypotheses using initial probabilities and experimental data (via the likelihood);
• the person who makes the inference -- or the ``user'' -- will make a decision for which he is fully responsible.

If one needs to compare two hypotheses, as in the example of the signal to noise calculation, the ratio of the final probabilities can be taken as a quantitative result of the test. Let us rewrite the $S/N$ formula () in the most general case:

$$\frac{P(H_1\,\vert\,E,H_\circ)}{P(H_2\,\vert\,E,H_\circ)} = \frac... ...\,H_2,H_\circ)} \cdot \frac{ P(H_1\,\vert\,H_\circ)}{P(H_2\,\vert\,H_\circ)}\,,$$ (3.24)

where again we have reminded ourselves of the existence of $H_\circ$. The ratio depends on the product of two terms: the ratio of the priors and the ratio of the likelihoods. When there is absolutely no reason for choosing between the two hypotheses the prior ratio is 1 and the decision depends only on the other term, called the Bayes factor. If one firmly believes in either hypothesis, the Bayes factor is of minor importance, unless it is zero or infinite (i.e. one and only one of the likelihoods is vanishing). Perhaps this is disappointing for those who expected objective certainty from a probability theory, but this is in the nature of things.