Sunday, January 09, 2022

Ruminating on Joint Probability vs Conditional Probability vs Marginal Probability

The concept of conditional probability is a very important to understand Bayesian networks. An excellent introduction to these concepts is available in this video - https://youtu.be/5s7XdGacztw

As we know, probability is calculated as the number of desired outcomes divided by the total possible outcomes. Hence if we roll a dice, the probability that it would be 4 is 1/6 ~ (P = 0.166 = 16.66%)

Siimilary, the probability of an event not occurring is called as the complement ~ (1-P). Hence the probability of not rolling a 4 would = 1-0.166 = 0.833 ~ 83.33%

While the above is true for a single variable, we also need to understand how to calculate the probability of two or more variables - e.g. probability of lightening and thunder happening together when it rains. 

When two or more variables are involved, then we have to consider 3 types of probability:

1) Joint probability calculates the likelihood of two events occurring together and at the same point in time. For example, the joint probability of event A and event B is written formally as:  P(A and B) or  P(A ^ B) or P(A, B)

2) Conditional probability measures the probability of one event given the occurrence of another event.  It is typically denoted as P(A given B) or P(A | B). For complex problems involving many variables, it is difficult to calculate joint probability of all possible permutations and combinations. Hence conditional probability becomes a useful and easy technique to solve such problems. Please check the video link above. 

3) Marginal probability is the probability of an event irrespective of the outcome of another variable.

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