(sec:Inference:statements)=
# 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:

$$ 
   \cprob{A}{B} \equiv \text{``probability of $A$ given $B$ is true''}.
$$

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. 

$$\cprob{\text{``String theory is true''}}{\text{``Data
from all high-energy colliders and cosmology''}},$$ 

or, less frivolously 

$$\cprob{\text{``I'll come to the party tonight''}}{\text{``You
invited me''}}.$$ 

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 {ref}`sec:BayesianEpistemology`.
:::


(sec:ManipulatingPDFs)=