Conditional probability table: Difference between revisions

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In [[statistics]], the '''conditional probability table (CPT)''' is defined for a set of discrete and mutually [[independence (probability)|dependent]] [[random variable]]s to demonstratedisplay [[conditional probability|conditional probabilities]] of a single variable with respect to the others (i.e., the probability of each possible value of one variable if we know the values taken on by the other variables). For example, assume there are three random variables <math>x_1,x_2, x_3</math> where each has <math>K</math> states. Then, the conditional probability table of <math>x_1</math> provides the conditional probability values <math>P(x_1=a_k\mid x_2,x_3)</math> – where the vertical bar <math>|</math> means “given the values of” – for each of the ''K'' possible values <math>a_k</math> of the variable <math>x_1</math> and for each possible combination of values of <math>x_2,\, x_3.</math> This table has <math>K^3</math> cells. In general, for <math>M</math> variables <math>x_1,x_2,\ldots,x_M</math> with <math>K_i</math> states for each variable <math>x_i,</math> the CPT for any one of them has the number of cells equal to the product <math>K_1K_2\cdots K_M.</math><ref name=murphybook>{{cite book|last=Murphy|first=KP|title=Machine learning: a probabilistic perspective|year=2012|publisher=The MIT Press}}</ref>

A conditional probability table can be put into [[matrix (mathematics)|matrix]] form. As an example with only two variables, the values of <math>P(x_1=a_k\mid x_2=b_j)=T_{kj},</math> with ''k'' and ''j'' ranging over ''K'' values, create a ''K''×''K'' matrix. This matrix is a [[stochastic matrix]] since the columns sum to 1; i.e. <math>\sum_k T_{kj} = 1</math> for all ''j''. For example, suppose that two [[binary variablesvariable]]s ''x'' and ''y'' have the [[joint probability distribution]] given in this table:

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With more than one conditioning variable, the table would still have one row for each potential value of the variable whose conditional probabilities are to be given, and there would be one column for each possible combination of values of the conditioning variables.

Moreover, the number of columns in the table could be substantially expanded to display the probabilities of the variable of interest conditional on specific values of only some, rather than all, of the other variables.

==References==

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[[Category:Conditional probability]]

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