Michael Data

Interpretation of Probability 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

Properties of Random Variables

Values of the variable are mutually exclusive and exhaustive.

Discrete

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.

Continuous

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.

# # # Sets of Random Variables

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. .

Conditional Distributions

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.

Factorization or Chain Rule

How do we get from a joint distribution to a continuous distribution? recursively: But order doesn't matter. Deriving Bayes' Rule

Factorization provides an easy derivation for Bayes' rule:
Let's represent .  Law of Total Probability

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.

Conditional Independence

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.

Genetics Example

G = maternal grandmother's genes
M = mother's genes
Y = your genes

Y and G are conditionally independent, given M.  Weather Example

I = Irvine temperature
B = Beijing temperature
M = month of year

I and B are conditionally independent, given M.

Linear Correlation This is a scaled covariance, and useful… 