Fiducial inference is one of a number of different types of statistical inference. These are rules, intended for general application, by which conclusions can be drawn from samples of data. In modern statistical practice, attempts to work with fiducial inference have fallen out of fashion in favour of frequentist inference, Bayesian inference and decision theory. However, fiducial inference is important in the history of statistics since its development led to the parallel development of concepts and tools in theoretical statistics that are widely used. Some current research in statistical methodology is either explicitly linked to fiducial inference or is closely connected to it.

The general approach of fiducial inference was proposed by Ronald Fisher. Here "fiducial" comes from the Latin for faith. Fiducial inference can be interpreted as an attempt to perform inverse probability without calling on prior probability distributions. Fiducial inference quickly attracted controversy and was never widely accepted. Indeed, counter-examples to the claims of Fisher for fiducial inference were soon published.^{[citation needed]} These counter-examples cast doubt on the coherence of "fiducial inference" as a system of statistical inference or inductive logic. Other studies showed that, where the steps of fiducial inference are said to lead to "fiducial probabilities" (or "fiducial distributions"), these probabilities lack the property of additivity, and so cannot constitute a probability measure.^{[citation needed]}

The concept of fiducial inference can be outlined by comparing its treatment of the problem of interval estimation in relation to other modes of statistical inference.

Fisher designed the fiducial method to meet perceived problems with the Bayesian approach, at a time when the frequentist approach had yet to be fully developed. Such problems related to the need to assign a prior distribution to the unknown values. The aim was to have a procedure, like the Bayesian method, whose results could still be given an inverse probability interpretation based on the actual data observed. The method proceeds by attempting to derive a "fiducial distribution", which is a measure of the degree of faith that can be put on any given value of the unknown parameter and is faithful to the data in the sense that the method uses all available information.

Unfortunately Fisher did not give a general definition of the fiducial method and he denied that the method could always be applied.^{[citation needed]} His only examples were for a single parameter; different generalisations have been given when there are several parameters. A relatively complete presentation of the fiducial approach to inference is given by Quenouille (1958), while Williams (1959) describes the application of fiducial analysis to the calibration problem (also known as "inverse regression") in regression analysis. Further discussion of fiducial inference is given by Kendall & Stuart (1973).

Fisher required the existence of a sufficient statistic for the fiducial method to apply. Suppose there is a single sufficient statistic for a single parameter. That is, suppose that the conditional distribution of the data given the statistic does not depend on the value of the parameter. For example, suppose that n independent observations are uniformly distributed on the interval ${\displaystyle [0,\omega ]}$. The maximum, X, of the n observations is a sufficient statistic for ω. If only X is recorded and the values of the remaining observations are forgotten, these remaining observations are equally likely to have had any values in the interval ${\displaystyle [0,X]}$. This statement does not depend on the value of ω. Then X contains all the available information about ω and the other observations could have given no further information.

The cumulative distribution function of X is

Probability statements about X/ω may be made. For example, given α, a value of a can be chosen with 0 < a < 1 such that