A priori (statistics)

A priori (statistics)

In statistics, a priori knowledge refers to prior knowledge about a population, rather than that estimated by recent observation. It is common in Bayesian inference to make inferences conditional upon this knowledge, and the integration of "a priori" knowledge is the central difference between the Bayesian and Frequentist approach to statistics. We need not be 100% certain about something before it can be considered "a priori" knowledge, but conducting estimation conditional upon assumptions for which there is little evidence should be avoided. "A priori" knowledge often consists of knowledge of the domain of a parameter (for example, that it is positive) that can be incorporated to improve an estimate. Within this domain the distribution is usually assumed to be uniform in order to take advantage of certain theoretical results (most importantly the central limit theorem).

Examples

Basic example

Suppose that we pick (without replacement) two red beads and three black beads from a bag; what is the probability that the next bead we pick out will be red? Without a priori knowledge, we cannot reasonably answer the question. But if we already knew that there were only two red beads in the bag, then we could be certain that the probability of picking out another red bead was in fact zero. In this instance, we are 100% certain that the probability is zero, essentially because the population is finite.

More theoretical example

Suppose that we are trying to estimate the coefficients of an autoregressive (AR) stochastic process based on recorded data, and we know beforehand that the process is stationary. Any AR(2) process is of the form::$X_\left\{k\right\} + heta_1 X_\left\{k-1\right\} + heta_2 X_\left\{k-2\right\} = epsilon_k$Under the classical frequentist approach, we would proceed with Maximum Likelihood Estimation (MLE), but instead we can integrate our knowledge into the Likelihood function and maximize our likelihood conditional upon the fact that the process is stationary. We can assign prior distributions to the AR coefficients $heta_1, heta_2$ that are uniform across a limited domain in line with the constraints upon stationary process coefficients. For an AR(2) process, the constraints are::$| heta_2| < 1,$:$heta_2 + 1 > | heta_1|$Adding this information will change the Likelihood function, and when we now use MLE to estimate the coefficients, we will in general obtain a better estimate. This is true in particular when we suspect that the coefficients are near the boundary of the stationary domain. Note that the distribution on the domain is uniform, so we have not made any assumptions about what the coefficients actually are, only their domain.

ee also

* Prior distribution
* Bayes' theorem

Wikimedia Foundation. 2010.

Look at other dictionaries:

• statistics weight — kvantinis statistinis svoris statusas T sritis Standartizacija ir metrologija apibrėžtis Kvantinio energijos lygmens multipletiškumas arba išsigimimas. atitikmenys: angl. statistics weight vok. statistisches Gewicht, n rus. статистический вес, m… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

• A priori — may refer to: * A priori (languages), a type of constructed language * A priori (statistics), a knowledge of the actual population * A priori and a posteriori (philosophy), used to distinguish two types of propositional knowledge *Apriori… …   Wikipedia

• A-priori-Verteilung — Die A priori Verteilung ist ein Begriff aus der bayesianischen Statistik. Inhaltsverzeichnis 1 Definition 2 (Nicht )informative A priori Verteilungen 3 Konjugierte A priori Verteilungen 4 Literatur …   Deutsch Wikipedia

• probability and statistics — ▪ mathematics Introduction       the branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation, and display of numerical data. Probability has its origin in the study of gambling… …   Universalium

• A priori probability — The term a priori probability is used in distinguishing the ways in which values for probabilities can be obtained. In particular, an a priori probability is derived purely by deductive reasoning. [Mood A.M., Graybill F.A., Boes D.C. (1974)… …   Wikipedia

• Non-parametric statistics — In statistics, the term non parametric statistics has at least two different meanings: The first meaning of non parametric covers techniques that do not rely on data belonging to any particular distribution. These include, among others:… …   Wikipedia

• bayesian statistics — a somewhat controversial statistical methodology that, unlike conventional statistics, which treats population parameters as fixed (though unknown) values, treats parameters as random variables with a specified probability distribution, termed… …   Medical dictionary

• probabilité a priori — kvantinis statistinis svoris statusas T sritis Standartizacija ir metrologija apibrėžtis Kvantinio energijos lygmens multipletiškumas arba išsigimimas. atitikmenys: angl. statistics weight vok. statistisches Gewicht, n rus. статистический вес, m… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

• Structured data analysis (statistics) — Structured data analysis is the statistical data analysis of structured data. Either in the form of a priori structure such as multiple choice questionnaires or in situations with the need to search for structure that fits the given data, either… …   Wikipedia