Frequentist inference

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.

While "Bayesian inference" is sometimes held to include the approach to inference leading to optimal decisions, a more restricted view is taken here for simplicity.

Basis

Frequentist inference has been associated with the frequentist interpretation of probability, specifically that any given experiment can be considered as one of an infinite sequence of possible repetitions of the same experiment, each capable of producing statistically independent results. In this view, the frequentist inference approach to drawing conclusions from data is effectively to require that the correct conclusion should be drawn with a given (high) probability, among this notional set of repetitions. However, exactly the same procedures can be developed under a subtly different formulation. This is one where a pre-experiment point of view is taken. It can be argued that the design of an experiment should include, before undertaking the experiment, decisions about exactly what steps will be taken to reach a conclusion from the data yet to be obtained. These steps can be specified by the scientist so that there is a high probability of reaching a correct decision where, in this case, the probability relates to a yet to occur set of random events and hence does not rely on the frequency interpretation of probability. This formulation has been discussed by Neyman, among others.

Similarly, Bayesian inference has often been thought of as almost equivalent to the Bayesian interpretation of probability and thus that the essential difference between frequentist inference and Bayesian inference is the same as the difference between the two interpretations of what a "probability" means. However, where appropriate, Bayesian inference (meaning in this case an application of Bayes' theorem) is used by those employing a frequentist interpretation of probabilities.

There are two major differences in the frequentist and Bayesian approaches to inference that are not included in the above consideration of the interpretation of probability:

• In a frequentist approach to inference, unknown parameters are often, but not always, treated as having fixed but unknown values that are not capable of being treated as random variates in any sense, and hence there is no way that probabilities can be associated with them. In contrast, a Bayesian approach to inference does allow probabilities to be associated with unknown parameters, where these probabilities can sometimes have a frequency probability interpretation as well as a Bayesian one. The Bayesian approach allows these probabilities to have an interpretation as representing the scientist's belief that given values of the parameter are true [see Bayesian probability - Personal probabilities and objective methods for constructing priors].
• While "probabilities" are involved in both approaches to inference, the probabilities are associated with different types of things. The result of a Bayesian approach can be a probability distribution for what is known about the parameters given the results of the experiment or study. The result of a frequentist approach is either a "true or false" conclusion from a significance test or a conclusion in the form that a given sample-derived confidence interval covers the true value: either of these conclusions has a given probability of being correct, where this probability has either a frequency probability interpretation or a pre-experiment interpretation.

See also

• Probability interpretations
• German tank problem