Bayes' theorem for uncertain quantities: derivation from a physicist's point of view

- Before doing the experiment we are uncertain of the values of and : we know neither the true value, nor the observed value. Generally speaking, this uncertainty is quantified by .
- Under the hypothesis that we observe , we can calculate
the conditional probability
- Usually we don't have , but this
can be calculated by
and :
- If we do an experiment we need to have a good idea of the behaviour of the apparatus; therefore must be a narrow distribution, and the most imprecise factor remains the knowledge about , quantified by , usually very broad. But it is all right that this should be so, because we want to learn about .
- Putting all the pieces together we get the standard formula
of Bayes' theorem for uncertain quantities: