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Difference between revisions of "A posteriori distribution"

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A conditional probability distribution of a random variable, to be contrasted with its unconditional or [[A priori distribution|a priori distribution]].
 
A conditional probability distribution of a random variable, to be contrasted with its unconditional or [[A priori distribution|a priori distribution]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100301.png" /> be a random parameter with an a priori density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100302.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100303.png" /> be a random result of observations and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100304.png" /> be the conditional density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100305.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100306.png" />; then the a posteriori distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100307.png" /> for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100308.png" />, according to the [[Bayes formula|Bayes formula]], has the density
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Let $\Theta$ be a random parameter with an a priori density $p(\theta)$, let $X$ be a random result of observations and let $p(x|\theta)$ be the conditional density of $X$ when $\Theta=\theta$; then the a posteriori distribution of $\Theta$ for a given $X=x$, according to the [[Bayes formula|Bayes formula]], has the density
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a0100309.png" /></td> </tr></table>
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$$p(\theta|x)=\frac{p(\theta)p(x|\theta)}{\int\limits_{-\infty}^\infty p(\theta)p(x|\theta)d\theta}.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003010.png" /> is a [[Sufficient statistic|sufficient statistic]] for the family of distributions with densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003011.png" />, then the a posteriori distribution depends not on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003012.png" /> itself, but on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003013.png" />. The asymptotic behaviour of the a posteriori distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003016.png" /> are the results of independent observations with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003017.png" />, is  "almost independent"  of the a priori distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010030/a01003018.png" />.
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If $T(x)$ is a [[Sufficient statistic|sufficient statistic]] for the family of distributions with densities $p(x|\theta)$, then the a posteriori distribution depends not on $x$ itself, but on $T(x)$. The asymptotic behaviour of the a posteriori distribution $p(\theta|x_1,\dots,x_n)$ as $n\to\infty$, where $x_j$ are the results of independent observations with density $p(x|\theta_0)$, is  "almost independent"  of the a priori distribution of $\Theta$.
  
 
For the role played by a posteriori distributions in the theory of statistical decisions, see [[Bayesian approach|Bayesian approach]].
 
For the role played by a posteriori distributions in the theory of statistical decisions, see [[Bayesian approach|Bayesian approach]].

Revision as of 13:46, 30 September 2014

A conditional probability distribution of a random variable, to be contrasted with its unconditional or a priori distribution.

Let $\Theta$ be a random parameter with an a priori density $p(\theta)$, let $X$ be a random result of observations and let $p(x|\theta)$ be the conditional density of $X$ when $\Theta=\theta$; then the a posteriori distribution of $\Theta$ for a given $X=x$, according to the Bayes formula, has the density

$$p(\theta|x)=\frac{p(\theta)p(x|\theta)}{\int\limits_{-\infty}^\infty p(\theta)p(x|\theta)d\theta}.$$

If $T(x)$ is a sufficient statistic for the family of distributions with densities $p(x|\theta)$, then the a posteriori distribution depends not on $x$ itself, but on $T(x)$. The asymptotic behaviour of the a posteriori distribution $p(\theta|x_1,\dots,x_n)$ as $n\to\infty$, where $x_j$ are the results of independent observations with density $p(x|\theta_0)$, is "almost independent" of the a priori distribution of $\Theta$.

For the role played by a posteriori distributions in the theory of statistical decisions, see Bayesian approach.

References

[1] S.N. Bernshtein, "Probability theory" , Moscow-Leningrad (1946) (In Russian)


Comments

References

[a1] E. Sverdrup, "Laws and chance variations" , 1 , North-Holland (1967) pp. 214ff
How to Cite This Entry:
A posteriori distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A_posteriori_distribution&oldid=33451
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article