Overview
Conditioning is the process of adjusting statistical calculations based on information that is given.
As a an example, suppose that two die are thrown, and you want to assess the probability that the sum of the die is less than six. Now, suppose you are given the information that one of the die landed on four. This new information will change your assessment of the sum of the two die.
Conditional Probability
Conditional probability is the assessed probability of an event after receiving information that some other event has occurred.
The probability that event {% A %} has occurred given that we know that event {% B %} has occurred is given by the following formula.
{% \mathbb{P}(A|B) = \frac{\mathbb{P} (A \cap B)}{\mathbb{P}(B)} %}
Bayes Rule is a formula for calculating conditional probabilities. It also forms the foundations of Bayesian statistics and Bayesian based machine learning.
Conditional Expectation
Standard (non-measure theoretic) conditional expectation is the expectation that results when conditional probabilities are used instead of the raw probabilities.
{% \mathbb{E}(Y|X=x) = \sum y \times p(y|x) %}