Empirical probability

Empirical probability

Empirical probability, also known as relative frequency, or experimental probability, is the ratio of the number favourable outcomes to the total number of trials [ [http://www.answers.com/topic/empirical-probability-statistics Empirical probability at answers.com] ] [Mood A.M., Graybill F.A., Boes D.C. (1974) Introduction to the Theory of Statistics (3rd Edition). McGraw-Hill. Section 2.3] , not in a sample space but in an actual sequence of experiments. In a more general sense, empirical probability estimates probabilities from experience and observation. [ [http://tpub.com/math2/90.htm Empirical probabilities at tpub.com] ] The phrase a posteriori probability has also been used an alternative to empirical probability or relative frequency. [Mood A.M., Graybill F.A., Boes D.C. (1974) Introduction to the Theory of Statistics (3rd Edition). McGraw-Hill. Sections 2.3] This unusual usage of the phrase is not directly related to Bayesian inference and not to be confused with its equally occasional use to refer to posterior probability, which is something else.

In statistical terms, the empirical probability is an estimate of a probability. If modelling using a binomial distribution is appropriate, it is the maximum likelihood estimate. It is the Bayesian estimate for the same case if certain assumptions are made for the prior distribution of the probability

An advantage of estimating probabilities using empirical probabilities is that this procedure is relatively free of assumptions. For example, consider estimating the probability among a population of men that they satisfy two conditions: (i) that they are over 6 feet in height; (ii) that they prefer strawberry jam to raspberry jam. A direct estimate could be found by counting the number of men who satisfy both conditions to give the empirical probability the combined condition. An alternative estimate could be found by multiplying the proportion of men who are over 6 feet in height with the proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two conditions are statistically independent.

A disadvantage in using empirical probabilities arises in estimating probabilities which are either very close to zero, or very close to one. In these cases very large sample sizes would be needed in order to estimate such probabilities to a good standard of relative accuracy. Here statistical models can help, depending on the context, and in general one can hope that such models would provide improvements in accuracy compared to empirical probabilities, provided that the assumptions involved actually do hold. For example, consider estimating the probability that the lowest of the daily-maximum temperatures at a site in February in any one year is less zero degrees Celsius. A record of such temperatures in past years could be used to estimate this probability. A model-based alternative would be to select of family of probability distributions and fit it to the dataset contain past yearly values: the fitted distribution would provide an alternative estimate of the required probability. This alternative method can provide an estimate of the probability even if all values in the record are greater than zero.

ee also

*Empirical distribution function
*Empirical measure
*Frequency probability
*A priori probability in relation to "a posteriori probabiliy"


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Empirical Probability — A form of probability that is based on some event occurring, which is calculated using collected empirical evidence. An empirical probability is closely related to the relative frequency in a given probability distribution. In order for a theory… …   Investment dictionary

  • Empirical distribution function — In statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability 1/ n at each of the n numbers in a sample.Let X 1,ldots,X n be iid random variables in mathbb{R} with the cdf F ( x… …   Wikipedia

  • Probability — is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the… …   Wikipedia

  • Empirical Bayes method — In statistics, empirical Bayes methods are a class of methods which use empirical data to evaluate / approximate the conditional probability distributions that arise from Bayes theorem. These methods allow one to estimate quantities… …   Wikipedia

  • Empirical measure — In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. Empirical measures are relevant to mathematical… …   Wikipedia

  • probability theory — Math., Statistics. the theory of analyzing and making statements concerning the probability of the occurrence of uncertain events. Cf. probability (def. 4). [1830 40] * * * Branch of mathematics that deals with analysis of random events.… …   Universalium

  • Empirical process — The study of empirical processes is a branch of mathematical statistics and a sub area of probability theory. It is a generalization of the central limit theorem for empirical measures. DefinitionIt is known that under certain conditions… …   Wikipedia

  • Probability interpretations — The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it… …   Wikipedia

  • probability — The mathematics of probability is well understood. Probability is a non negative, additive set function whose maximum value is unity. What is harder to understand is the application of the formal notion to the actual world. One point of… …   Philosophy dictionary

  • 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

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”