Conditional Probabilities in Artificial Intelligence

Conditional Probabilities

In probability theory, a condition probability of A_i given A_j is is the probability of A_i if A_i and A_j is known to occur (or have occurred).

It is commonly denoted by P(A_i | A_j) and sometimes P_Aj(A_i).

${\rm{P(}}{{\rm{A}}_i}{\rm{ | }}{{\rm{A}}_j}){\rm{ = }}\frac{{P({A_i},{\rm{ }}{{\rm{A}}_j})}}{{P({A_j})}}$

Here P(A_i, A_j) is the joint probability of A_i and A_j and also denoted as P(A_i ∩ A_j) and P(Aj) is the marginal probability of A_j.

The joint probability can also be represented in the terms of conditional, probability as follows,

P(A_i, A_j) = P(A_i | A_j) P(A_j)

Consider an example where we can calculate the probability that the student has failed if he does not study well.

$P(F = True,{\rm{ }}S = False){\rm{ }} = {\rm{ }}\frac{{P(F = True,{\rm{ }}S = False)}}{{P(S = False)}}$

A conditional probability represents the normalized version of joint probability.

The conditional and joint probabilities can be explained by using the venn diagrams.

Let us take the same example of a student studies or not and if he fails or not, P(F, ᆨS) is represented as the overlapping region of the ellipses in the below Figure. 3.2.1.

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Fig 3.2.1: Venn Diagram

It denotes F = true and S = false. The region outside the ellipses denotes the condition where the student studies and does not fail. The region where S = false denotes that the student does not study and the region F = true denote the student fails.

NOTE 1: The marginal probabilities can be calculated from the joint probabilities from the above Figure. 3.2.1 as follows,

P(F) = P(F, S) + P(F, ᆨS)

NOTE 2: From several variables conditioned on other variables we can have joint conditional probabilities.

Example: ${\rm{P(}}\neg {\rm{Q, R | }}\neg {\rm{S, T) = }}\frac{{{\rm{P(}}\neg {\rm{Q, R, }}\neg {\rm{Q, T)}}}}{{P(\neg S,{\rm{ T)}}}}$

NOTE 3: A joint probability can be expressed in terms of a chain conditional probabilities.

Example : P(Q, R, S, T) = P(Q | R, S, T) P(R | S, T) P(S | T) P (T)

This chain rule has it general form,

$P({A_1},{\rm{ }}{{\rm{A}}_2},.....{A_k}){\rm{ = }}\prod\limits_{i = 1}^k {P(A{}_i{\rm{ |}}} {\rm{ }}{{\rm{A}}_{i - 1}},....{A_1})$

The way of ordering of variables in a joint probability function is not so important. Therefore the joint probability function can be written as,

$P({A_i},{\rm{ }}{A_j}){\rm{ = P(}}{{\rm{A}}_i}{\rm{ | }}{{\rm{A}}_j}){\rm{ P(}}{{\rm{A}}_j}){\rm{ = P(}}{{\rm{A}}_j}{\rm{ | }}{{\rm{A}}_i}){\rm{ P(}}{{\rm{A}}_i}){\rm{ = }}P({A_j},{\rm{ }}{A_i})$

It can be noted that,

$P({A_i},{\rm{ }}{A_j}){\rm{ = }}\frac{{{\rm{P(}}{{\rm{A}}_j}{\rm{ | }}{{\rm{A}}_i})}}{{{\rm{P(}}{{\rm{A}}_j})}}$

The above equation is known as Bayes rule.

p(v) is used as an abbreviation for P(A₁, A₂, ….A_k) where v = {A₁, A₂, ….A_k}

Conditional Probabilities in Artificial Intelligence