4.1. Statements#
A Bayesian approach to the world assigns probabilities to truth claims. It also recognizes that the probability of claim \(A\) is contingent upon statement \(B\) being true. We therefore introduce notation for conditional probability:
For a Bayesian \(A\) and \(B\) could stand for anything. But, whatever they are, statements can now be categorized as definitively true (probability 1), or definitively false (probability 0), or anything in between.
Examples:
\(\cprob{\text{``Betty eats grass''}}{\text{``Betty is a cow'' and ``All cows eat grass''}}=1\)
\(\cprob{\text{``Lines A and B intersect''}}{\text{``A and B are parallel lines in flat space''}}=0\)
\(\cprob{\text{``Particle A is negatively charged''}}{\text{``A is an electron''}}=1\)
On the other hand something like \(\cprob{\text{``below 0 deg,. C''}}{\text{``it is January in Ohio''}}\) should be assigned a probability between 0 and 1. We might even estimate this probability based on our past experience of Januarys in Ohio. But even statements for which we cannot construct a \(\text{``frequentist estimator''}\) can be assigned probabilities, e.g.
or, less frivolously
This emphasizes that, for a Bayesian, the probability is interpreted as representative of the Bayesian’s current state of knowledge, and not necessarily as the long-term average of a set of many trials. After all, what would it mean to conduct a set of trials, in some of which string theory was true, and in some of which it was not?
Warning
It is easy to forget the meaning of the key word given and assume that \(\cprob{A}{B}\) means that we know \(B\) is actually true, which may or may not be the case.
Note
Further discussion on the interpretation of probability can be found in *Aside: Bayesian epistemology.