# A priori distribution

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The probability distribution of a random variable, to be contrasted with the conditional distribution of this random variable under certain additional conditions. Usually the term "a priori distribution" is used in the following way. Let $(\Theta,X)$ be a pair of random variables (random vectors or more general random elements). The random variable $\Theta$ is considered to be unknown, while $X$ is considered to be the result of an observation to be used for estimation of $\Theta$. The joint distribution of $\Theta$ and $X$ is given by the distribution of $\Theta$ (now called the a priori distribution) and the set of conditional probabilities $\mathrm P_\theta$ of the random variable $X$ given $\Theta=\theta$. According to the Bayes formula, one can calculate the conditional probability of $\Theta$ with respect to $X$ (which is now called the a posteriori distribution of $\Theta$). In statistical problems, the a priori distribution is often unknown (and even the assumption on its existence is not sufficiently founded). For the use of the a priori distribution, see Bayesian approach.

#### References

 [a1] E. Sverdrup, "Laws and chance variations" , 1 , North-Holland (1967) pp. 214ff
How to Cite This Entry:
A priori distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A_priori_distribution&oldid=43646
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article