or

* is a proposition that is true or false

* Frequentist or Bayesian interpretation of the value?

** Frequentist: $P(a)$ is the relative frequency with which $a$ occurs in repeated trials
** Bayesian: is a degree of belief of an “agent” that proposition is true

Values of the variable are mutually exclusive and exhaustive.

The variable takes discrete values. . Discrete distributions can be represented as a table of probabilities for each value. Alternately, the distribution can be represented by some function of the values.

The variable takes continuous values. . The distribution is typically represented with a density function. There is no real bound on the height of the density curve.

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e.g. consider discrete random variables A,B,C. Each takes m different values, with .

is a **joint distribution**. It can be represented as a table with probabilities for each permutation of values. .

e.g. can now be represented as a table of values, because the values of and have been fixed. The conditioned variables are either known or assumed to have those values. .

If we were to plot across different values of , the result is not a density function and does not have to integrate or sum to 1.

How do we get from a joint distribution to a continuous distribution?

recursively:

But order doesn't matter.

Factorization provides an easy derivation for Bayes' rule:

Let's represent .

How can we get from a joint distribution to **unconditional** a.k.a. **marginal** distributions?

For discrete random variables, sum out the extra variables:

For continuous random variables, integrate out the extra variables.

A **conditional independence** assumption allows reduction of the amount of information required when storing or working with a joint distribution.

e.g. for discrete random variables A,B,C,D which each take m values:

By factorization,

This uses at least terms!

We can assume that A is **conditionally independent** of C and D given B:

Now we factorize like this:

Now the largest table of the factorized representation is terms.

G = maternal grandmother's genes

M = mother's genes

Y = your genes

Y and G are conditionally independent, given M.

I = Irvine temperature

B = Beijing temperature

M = month of year

I and B are conditionally independent, given M.

This is a scaled covariance, and useful…