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The probability distribution of a random variable, to be contrasted with the [[Conditional distribution|conditional distribution]] of this random variable under certain additional conditions. Usually the term "a priori distribution" is used in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100501.png" /> be a pair of random variables (random vectors or more general random elements). The random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100502.png" /> is considered to be unknown, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100503.png" /> is considered to be the result of an observation to be used for estimation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100504.png" />. The joint distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100506.png" /> is given by the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100507.png" /> (now called the a priori distribution) and the set of conditional probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100508.png" /> of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a0100509.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a01005010.png" />. According to the [[Bayes formula|Bayes formula]], one can calculate the conditional probability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a01005011.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a01005012.png" /> (which is now called the a posteriori distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010050/a01005013.png" />). 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|Bayesian approach]].
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The [[probability distribution]] of a random variable, to be contrasted with the [[Conditional distribution|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|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]].
 
 
 
 
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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Sverdrup,  "Laws and chance variations" , '''1''' , North-Holland  (1967)  pp. 214ff</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Sverdrup,  "Laws and chance variations" , '''1''' , North-Holland  (1967)  pp. 214ff</TD></TR>
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Latest revision as of 19:22, 21 April 2024

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=12611
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article