What does it mean to factor a joint distribution? A book I'm reading (Hogan & Mason, 2012, p37) contains the following passage:

The joint distribution can be factored in two different ways into
  conditional and marginal probabilities that reveal different aspects
  of forecast quality. The calibration-refinement factorization is given by
  $p\left ( \hat{x},x \right) = p\left (x\mid \hat{x}\right)p\left (\hat{x}\right)$
  [...]
  The second way of factoring the joint distribution is known as the likelihood-base rate factorization and is given by
  $p\left ( \hat{x},x \right) = p\left (\hat{x}\mid x\right)p\left (x\right)$

The context for this passage is the discussion of 2 × 2 contingency tables, where the rows represent binary forecasts and the columns represent binary outcomes. On p32 of the same book a schematic table of this sort is provided.

Ideally I'd hope for an answer that doesn't just provide a correct definition, and which also helps to build intuitions about what factorization means and what the point of it is. 
Hogan, R. J., & Mason, I. B. (2012). Deterministic forecasts of binary events. Forecast Verification: A Practitioner's Guide in Atmospheric Science, Second Edition, 31-59.
 A: If I am understanding the passage and table correctly, there are essentially two answers: mathematical and qualitative.
Qualitatively, the "different aspects of forecast quality" is essentially the idea of false positives vs. false negatives, or precision vs. recall. (I will come back to this.)
Mathematically, the table is comparing two binary variables
$$F =\text{Was an event forecast?}\quad\text{vs.}\quad O =\text{Did an event occur?}$$
and tallying their co-occurences.
The empirical joint distribution of these variables would be
$$
\begin{bmatrix}
p(\phantom{\sim}F,O) & p(\phantom{\sim}F,\sim\!O) \\
p(\sim\!F,O) & p(\sim\!F,\sim\!O) \\
\end{bmatrix}
=
\frac{1}{n}
\begin{bmatrix}
a & b \\
c & d \\
\end{bmatrix}
$$
where
$$\sim\!F=\text{not }F \quad,\quad \sim\!O=\text{not }O$$
and $n=a+b+c+d$.
The marginal probabilities appear in the margins of the table, and give the total probabilities for forecasts (row sums)
\begin{align}
p(\phantom{\sim}F) &= p(\phantom{\sim}F,O) + p(\phantom{\sim}F,\sim\!O) = \frac{a+b}{n} \\
p(\sim\!F) &= p(\sim\!F,O) + p(\sim\!F,\sim\!O) = \frac{c+d}{n}
\end{align}
and observations (column sums)
\begin{align}
p(\phantom{\sim}O) &= p(F,\phantom{\sim}O) + p(\sim\!F,\phantom{\sim}O) = \frac{a+c}{n} \\
p(\sim\!O) &= p(F,\sim\!O) + p(\sim\!F,\sim\!O) = \frac{b+d}{n} \\
\end{align}
What is not shown in the table are the conditional probabilities. For example what is the probability of an event given that one was forecast? This is given by
$$p(O\mid F)=\frac{p(F,O)}{p(F)}=\frac{a}{a+b}$$
(which could be called the precision). The complement of this would be the fraction of false alarms $p(\sim\!O\mid F)$ (or false positives, more generally).
Another perspective would be to condition on $O$. For example the fraction of events that were forecast is given by
$$p(F\mid O)=\frac{p(F,O)}{p(O)}=\frac{a}{a+c}$$
(which could be called the recall). The complement of this would be the fraction of false negatives $p(\sim\!F\mid O)$.
Mathematically, these perspectives correspond to two different ways of factoring the joint probability
$$p(O\mid F)\,p(F)=p(F,O)=p(F\mid O)\,p(O)$$
(Due to this, a punny personality such as myself might describe the joint probability as the "missing link" in Bayes theorem!)
