# Is there a name for models that are based on a given conditional distribution only?

Suppose I have the conditional probabilities of $B$ given $A$ (both categorical): $$\Pr(B|A).$$ The probabilities are given as a contingency table.

Using these conditional probabilities, I can fudge a $B$ column into a dataset that has only an $A$ column. For each row in the dataset, I choose a value for $B$ at random based on $P(B|A)$ for the value of $A$ of the current row.

From my understanding, this can be seen as a model, although a quite simple one. Adding the $B$ column could be thought of as "simulating" the model.

My question: Is the term "model" applicable in this case? If so, how would such a model be called?

Somebody mentioned the term "Poisson distribution" in this context, but I was unable to align these two concepts with my limited knowledge of statistics.

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I would call it a multinomial model. For each possible value of $A$, $B$ follows a different multinomial distribution. But that doesn't really add anything except jargon. You might say more about the nature of $A$, $B$ and $\Pr(B | A)$.
$A$ and $B$ are both categorical (discrete) variables. $\Pr(B|A)$ is given as map from $A \times B$ to $[0, 1]$ and is otherwise unspecified. – krlmlr Sep 25 '11 at 6:37
Suppose I have a large sample with an $A$ attribute and a smaller, but still representative, sample with $A$ and $B$ attributes. I want to add a $B$ attribute to the large sample using the conditional probabilities suggested by the small sample and the values of the $A$ column in the large sample. – krlmlr Sep 26 '11 at 22:20
@user946850 - There's no real name for this model that is more precise than multinomial (which covers any discrete distribution), unless $\Pr(B|A)$ had a particular form. Your particular would be called imputation. The Poisson distribution is a particular discrete distribution that could match $\Pr(B|A)$ but probably is not at all related to this. – Karl Sep 27 '11 at 3:12