## Fundamentals

A random variable is denoted in capital, and the values it can take is denoted in small .

Consider a collection of random variables . Random variables can be thought of as features of a particular domain of interest.

For example, the result of a coin toss can be represented using a single random variable, . This variable can take either of the categorical values or . If the same coin is tossed times, this can be represented using variables . Each of these values can be either or .

## Joint Probability

An expression of the form is called a joint probability function over the variables . The *joint probability* is defined when the values of are respectively. This is denoted by the expression

This is sometimes abbreviated as .

For a fair coin toss, . If a fair coin is tossed five times, .

The joint probability function satisfies

## Marginal Probability

The *marginal probability* of one of the random variables can be computed if the values of all of the joint probabilities for a set of random variables are known.

For example, the marginal probability is defined to be the sum of all those joint probabilities for which

When dealing with propositional variables (True/False) TrueFalse is denoted as .

## Conditional Probabilities

The conditional probability of given is denoted by .

where is the joint probability of and and is the marginal probability of . Thus

Joint conditional probabilities of several variables conditioned on several other variables is expressed as

A joint probability can be expressed in terms of a *chain* of conditional probabilities.

The general form of this *chain* rule is

## Bayes Rule

Different possible orders give different expressions but they all have the same value for the same set of variable values. Since the order of variables is not important

Which gives *Bayes’ Rule*

## Probabilistic Inference

In set notation , , The variables having the values respectively is denoted by , where and are ordered lists.

For a set , the variables in a subset of are given as *evidence*.

For example, consider . The evidence is being false. In other words equates to .

Thus need not be computed.

## Conditional Independence

A variable is conditionally independent of a set of variables given a set if

(1)

tells nothing more about than is already known by knowing .

(2)

Saying that is conditionally independent of given also means that is conditionally independent of given . The same result also applies to sets and .

As a generalization of pairwise independence, the variables are *mutually conditionally independent*, given a set if each of the variables is conditionally independent of all of the others given .

(3)

When is empty

This implies that the variables are unconditionally independent.

### Thank You

**Mark**– for pointing out typos and errors

Another great article! Nice, clear explanations.

I do have some suggestions for improvement though:

It seems there is an error in the Bayes Rule section. It should be P(V_i|V_j) = P(V_j|V_i)P(V_i)/P(V_j)

I would also recommend explaining how you introduced the P and not P random variables in the Probabilistic Inference section as it is not quite clear how that works out.

Finally, it seems your Latex implementation had trouble rendering for the fourth paragraph in the Conditional Independence section (\textbf{mutually conditionally independent}), right above equation (3).

I have expanded the Probabilistic Inference section and fixed the errors you have pointed out. Thank you.