Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.
Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials. Probabilities can be found by a repeatable objective process. This interpretation supports the statistical needs of many experimental scientists and pollsters. It does not support all needs, however; gamblers typically require estimates of the odds without experiments.
There are several kinds of mean in mathematics, especially in statistics.
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.
Statistics is the discipline that concerns the collection, organization, analysis, interpretation and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics.
Statistics is a field of inquiry that studies the collection, analysis, interpretation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the physical and social sciences to the humanities; it is also used and misused for making informed decisions in all areas of business and government.
A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables. A statistical hypothesis test is a method of statistical inference. Commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model. An alternative hypothesis is proposed for the statistical-relationship between the two data-sets, and is compared to an idealized null hypothesis that proposes no relationship between these two data-sets. This comparison is deemed statistically significant if the relationship between the data-sets would be an unlikely realization of the null hypothesis according to a threshold probability—the significance level. Hypothesis tests are used when determining what outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified level of significance.
In statistics, the likelihood function measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values.
In probability theory and statistics, Bayes' theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if the risk of developing health problems is known to increase with age, Bayes’s theorem allows the risk to an individual of a known age to be assessed more accurately than simply assuming that the individual is typical of the population as a whole.
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a sequence of data. Bayesian inference has found application in a wide range of activities, including science, engineering, philosophy, medicine, sport, and law. In the philosophy of decision theory, Bayesian inference is closely related to subjective probability, often called "Bayesian probability".
Odds are a numerical expression, usually expressed as a pair of numbers, used in both gambling and statistics. In statistics, the odds for or odds of some event reflect the likelihood that the event will take place, while odds against reflect the likelihood that it will not. In gambling, the odds are the ratio of payoff to stake, and do not necessarily reflect exactly the probabilities. Odds are expressed in several ways, and sometimes the term is used incorrectly to mean simply the probability of an event. Conventionally, gambling odds are expressed in the form "X to Y", where X and Y are numbers, and it is implied that the odds are odds against the event on which the gambler is considering wagering. In both gambling and statistics, the 'odds' are a numerical expression of the likelihood of some possible event.
In statistics, a confidence interval (CI) is a type of estimate computed from the statistics of the observed data. This proposes a range of plausible values for an unknown parameter. The interval has an associated confidence level that the true parameter is in the proposed range. Given observations and a confidence level , a valid confidence interval has a probability of containing the true underlying parameter. The level of confidence can be chosen by the investigator. In general terms, a confidence interval for an unknown parameter is based on sampling the distribution of a corresponding estimator.
Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a degree of belief in an event. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other interpretations of probability, such as the frequentist interpretation that views probability as the limit of the relative frequency of an event after many trials.
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.
The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the development, dissemination, and application of statistics and probability. The Institute currently has about 4,000 members in all parts of the world. Beginning in 2005, the institute started offering joint membership with the Bernoulli Society for Mathematical Statistics and Probability as well as with the International Statistical Institute. The Institute was founded in 1935 with Harry C. Carver and Henry L. Rietz as its two most important supporters. The institute publishes a variety of journals, and holds several international conference every year.
Frequentist inference is a type of statistical inference that draws conclusions from sample data by emphasizing the frequency or proportion of the data. An alternative name is frequentist statistics. This is the inference framework in which the well-established methodologies of statistical hypothesis testing and confidence intervals are based. Other than frequentistic inference, the main alternative approach to statistical inference is Bayesian inference, while another is fiducial inference.
