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Frequentist inference

Frequentist inference is a foundational paradigm in statistics that interprets probability as the long-run relative frequency of events occurring in an infinite sequence of repeated random experiments under identical conditions, enabling inferences about fixed but unknown population parameters from observed sample data.^[1] This approach focuses on developing procedures with controlled long-run error rates, such as the probability of Type I errors in hypothesis testing, without assigning probabilities directly to the parameters themselves.^[2] Unlike Bayesian methods, which incorporate prior beliefs and update them with data to yield posterior probabilities for parameters, frequentist inference treats parameters as deterministic constants and emphasizes the behavior of statistical procedures over hypothetical repetitions of the experiment.^[3] The development of frequentist inference is closely associated with the work of Ronald A. Fisher, Jerzy Neyman, and Egon S. Pearson in the early 20th century.^[4] Fisher laid early groundwork through his emphasis on randomization in experimental design and the use of significance tests to assess evidence against a null hypothesis, as detailed in his influential 1925 book Statistical Methods for Research Workers, which introduced concepts like the p-value as a measure of the strength of evidence.^[5] Neyman and Pearson extended this framework in their 1933 paper "On the Problem of the Most Efficient Tests of Statistical Hypotheses," where they formalized hypothesis testing as a decision-theoretic process, defining power functions and optimal tests that balance Type I and Type II error rates under alternative hypotheses. Neyman further advanced the theory in 1937 with his concept of confidence intervals, which provide a range of plausible values for a parameter such that, over repeated sampling, the interval contains the true parameter with a specified probability (e.g., 95%).^[6] Central tools in frequentist inference include null hypothesis significance testing (NHST), confidence intervals, and point estimation methods like maximum likelihood estimation.^[7] In NHST, a null hypothesis (often denoting no effect or difference) is tested against data, with rejection based on a p-value below a pre-set significance level (typically 0.05), controlling the long-run false positive rate.^[1] Confidence intervals complement this by quantifying uncertainty around estimates, while point estimators aim for properties like unbiasedness and minimum variance, as evaluated through criteria such as the Neyman-Pearson lemma for optimality. These methods underpin much of modern applied statistics in fields like medicine, economics, and physics, where they facilitate decision-making under uncertainty by guaranteeing procedure performance in repeated use.^[4]

Foundations

Core Definition

Frequentist inference constitutes a foundational framework in statistics wherein probability is construed as the limiting relative frequency of an event occurring in an infinite sequence of repeated trials conducted under identical conditions.^[8] This interpretation underpins all probabilistic statements, emphasizing empirical long-run frequencies rather than subjective beliefs, and forms the basis for deriving inference procedures from observable data without invoking prior distributions.^[8] Within this paradigm, population parameters—such as means or proportions—are regarded as fixed, unknown constants that do not possess probability distributions of their own.^[6] In contrast, the observed data are treated as realizations of random variables, with variability arising solely from the sampling process under the fixed parameter values.^[9] This dichotomy ensures that uncertainty is quantified through the randomness in the data, enabling objective assessments of parameter values via repeated hypothetical sampling. Central to frequentist inference are pivotal quantities, which are functions of both the data and the unknown parameters whose probability distributions remain invariant to the specific value of the parameter.^[6] These pivots facilitate inference by allowing the construction of intervals or tests with known coverage probabilities, independent of priors. For instance, consider a pivotal quantity g(\theta, X) with a distribution known unconditionally; the corresponding (1 - \alpha) confidence interval for the parameter \theta is the set \{\theta : c_1 \leq g(\theta, X) \leq c_2\}, where c_1 and c_2 satisfy P(c_1 \leq g(\theta, X) \leq c_2) = 1 - \alpha for all \theta.^[6] Frequentist approaches delineate point estimation, which yields a single numerical approximation for the parameter (e.g., the sample mean as an estimate of the population mean), from interval estimation, which delivers a range of values incorporating uncertainty through confidence intervals that guarantee a specified long-run coverage rate across repeated experiments.^[9] While point estimates prioritize simplicity and bias reduction, interval estimates emphasize reliability by quantifying the precision of the inference.^[9]

Frequentist Probability

In the frequentist interpretation, probability is defined as the limiting relative frequency of an event in an infinite sequence of repeatable trials under identical conditions. Specifically, for a given event A, the probability P(A) is the limit \lim_{n \to \infty} \frac{m_n}{n}, where n is the number of trials and m_n is the number of occurrences of A in those trials.^[10] This objective measure relies on the assumption that the experiment can be repeated indefinitely, allowing the observed frequency to converge to a stable value that reflects the underlying chance mechanism.^[11] This view contrasts sharply with subjective or axiomatic interpretations of probability, such as those in Bayesian statistics, where probabilities represent degrees of belief updated via priors and likelihoods. Frequentist probability eschews informative priors, treating probabilities as fixed properties of the world discoverable through long-run frequencies rather than personal judgments.^[11] In frequentism, there are no non-informative priors in the Bayesian sense; instead, uncertainty is quantified solely through the variability in repeated sampling, assuming fixed but unknown parameters.^[12] Richard von Mises formalized this frequency approach through two key axioms for defining random sequences, or "collectives," which are infinite sequences of trial outcomes exhibiting stable frequencies. The axiom of convergence requires that the relative frequency of any attribute (event) in the sequence approaches a definite limit as the number of trials increases to infinity.^[13] The axiom of randomness stipulates that this limiting frequency remains unchanged in every infinite subsequence obtained by a place-selection rule—one that depends only on the order of previous outcomes, ensuring no systematic bias in subsequence choice.^[13] These axioms ensure that probabilities are invariant and empirically grounded, avoiding ad hoc adjustments to sequences.^[14] A classic example of probability assignment under this framework is the Bernoulli trial, such as repeated fair coin flips, where each trial has two outcomes (heads or tails) with fixed probabilities p and $1-p. For a fair coin, p = 1/2, so the probability of heads is the long-run proportion of heads observed over infinitely many flips, converging to 0.5.^[15] In sampling contexts, this extends to assigning probabilities to outcomes in random samples from a population, such as drawing balls from an urn with replacement, where the probability of selecting a specific color stabilizes as the limiting frequency in repeated draws.^[15] Frequentist probability plays a central role in defining sampling distributions, which describe the probability distribution of a statistic computed from random samples of fixed size drawn from a population. For instance, the sampling distribution of the sample mean \bar{X} for independent identically distributed observations from a population with mean \mu and variance \sigma^2 is centered at \mu with variance \sigma^2 / n, where n is the sample size; as n grows, this distribution often approximates a normal distribution by the central limit theorem, enabling probabilistic statements about the statistic's behavior across repeated samples.^[11] This foundation supports frequentist inference by providing the long-run frequency basis for assessing statistic variability under fixed parameters.^[11]

Historical Development

Early Foundations

The foundations of frequentist inference trace back to the early 18th century with Jacob Bernoulli's formulation of the weak law of large numbers in his 1713 work Ars Conjectandi. This theorem demonstrated that, for a sequence of independent Bernoulli trials with fixed success probability, the sample proportion converges in probability to the true probability as the number of trials increases, establishing a mathematical basis for viewing probabilities as limiting frequencies in repeated experiments.^[16] Bernoulli's result justified the use of observed frequencies to estimate underlying probabilities, shifting emphasis toward empirical long-run behavior rather than subjective degrees of belief.^[17] In the early 19th century, Siméon Denis Poisson built upon Bernoulli's ideas in his 1837 treatise Recherches sur la Probabilité des Jugements en Matière Criminelle et en Matière Civile, where he formalized the law of large numbers and explored its implications for probability limits in legal and social contexts. Poisson showed that the relative frequency of events stabilizes around their expected probabilities under repeated observations, providing tools for assessing the reliability of judgments based on aggregate data.^[18] Concurrently, Adolphe Quetelet applied these principles to social statistics in works such as Sur l'homme et le développement de ses facultés, ou Essai de physique sociale (1835), demonstrating that phenomena like crime rates and birth ratios exhibited predictable regularities when examined across large populations, akin to physical laws. Quetelet's "social physics" used the law of large numbers to argue that individual variations average out, revealing underlying deterministic patterns in human behavior.^[19] Pierre-Simon Laplace advanced these developments through his principle of inverse probability, outlined in Théorie Analytique des Probabilités (1812), where he approximated posterior distributions using uniform priors and normal error assumptions, leading to methods that prefigured least squares estimation. Laplace's approximations justified treating errors as normally distributed and enabled probabilistic inferences from data without explicit Bayesian priors, though still rooted in inverse reasoning.^[20] Carl Friedrich Gauss contributed significantly to error theory in astronomy with his 1809 publication Theoria Motus Corporum Coelestium, deriving the normal distribution as the probability density that minimizes the expected squared error for observational discrepancies. Gauss's approach assumed errors arise from numerous small, equally likely causes and established least squares as the optimal method for parameter estimation under this model, emphasizing direct probability statements about error distributions rather than parameters themselves.^[21] By the mid-19th century, these contributions facilitated a transition from inverse probability methods—often seen as proto-Bayesian due to their focus on updating parameter beliefs—to direct probability approaches that prioritized frequency-based statements about observable quantities like errors and test statistics. This shift, evident in the growing application of normal approximations and least squares to empirical data in astronomy and social sciences, laid the groundwork for modern frequentist inference by centering on long-run frequencies and sampling distributions.^[17]

Key Formulations in the 20th Century

In the 1920s, Ronald A. Fisher developed foundational methods for frequentist inference, introducing maximum likelihood estimation as a principle for selecting parameter values that maximize the probability of observed data under a statistical model.^[22] This approach, detailed in his 1922 paper, emphasized the likelihood function as a tool for point estimation without relying on prior distributions, marking a shift toward objective inference based on data alone.^[22] Fisher also advanced significance testing through the concept of p-values, which quantify the probability of observing data as extreme as or more extreme than the sample under the null hypothesis, as outlined in his 1925 book where he recommended a 5% threshold for assessing evidence against the null.^[23] A pivotal advancement came in 1933 with the Neyman-Pearson lemma, which provided a framework for constructing optimal tests of simple hypotheses by maximizing power while controlling the test's size.^[24] The lemma specifies that, for testing a null hypothesis H_0: \theta = \theta_0 against an alternative H_1: \theta = \theta_1, the most powerful test rejects H_0 if the likelihood ratio \Lambda = \frac{L(\theta_0)}{L(\theta_1)} < k, where k is chosen to ensure the test size \alpha = P(\Lambda < k \mid H_0) does not exceed a predetermined level.^[24] This formulation introduced the power function \beta(\theta) = 1 - P(\text{reject } H_0 \mid \theta), which measures the probability of correctly rejecting the null when it is false, thus balancing error control in hypothesis testing. Neyman extended this framework in 1937 by introducing confidence intervals, a method for constructing ranges of plausible values for unknown parameters such that the interval contains the true value with a specified coverage probability (e.g., 95%) over repeated sampling from the same population.^[6]^[24] Fisher extended his ideas in 1930 with fiducial inference, proposing a method to derive a probability distribution for unknown parameters directly from the sampling distribution of a pivotal quantity, treating the parameter as a random variable in a "fiducial" sense. This approach aimed to provide interval estimates analogous to confidence intervals but rooted in the fiducial probability statement, influencing later developments in interval estimation despite ongoing debates about its logical foundations.^[25] Tensions in these formulations surfaced in 1935 through correspondence and exchanges between Fisher and Jerzy Neyman, particularly following Neyman's presentation on agricultural experimentation, where they debated the goals of inference—Fisher emphasizing inductive reasoning via p-values for scientific discovery, while Neyman advocated behavioristic decision-making focused on long-run error rates.^[26] By the 1940s, these ideas evolved into unified frequentist frameworks, incorporating type I error rate \alpha (probability of false rejection of the null) and type II error rate \beta (probability of false acceptance), as Neyman and Egon Pearson refined their theory to encompass composite hypotheses and estimation procedures.^[27] This synthesis, building on the 1933 lemma, established error-based criteria for test selection, solidifying frequentist inference as a decision-theoretic paradigm.

Philosophical Underpinnings

Core Principles of Frequentism

Frequentist inference rests on the principle of long-run frequency, wherein probability is interpreted as the limiting relative frequency of an event in an infinite sequence of independent repetitions under identical conditions. This approach validates inferences by considering their reliability over hypothetical repeated sampling from the same population, rather than assessing the probability of a specific observed outcome or parameter value in isolation.^[8] Inferences are thus deemed valid if the procedure yields correct conclusions with a specified frequency in the long run, emphasizing repeatability and empirical stability over singular events.^[6] A cornerstone of frequentism is its commitment to objectivity, achieved by excluding subjective prior beliefs and relying solely on evidence derived from the observed data and the sampling process. Unlike approaches that incorporate personal judgments, frequentist methods calibrate inferences using the sampling distribution of statistics, ensuring that conclusions connect directly to the data-generating mechanism without preconceived notions.^[28] This focus on data-driven evidence positions the statistician as a guardian of objectivity, quantifying potential errors through frequencies observable in repeated experiments.^[28] Frequentism rejects the assignment of probabilities to parameters, treating them as fixed but unknown constants rather than random variables. Consequently, expressions like P(\theta \in C), where \theta is a parameter and C an interval, are undefined within this framework, as probability applies only to observable random variables subject to long-run frequencies.^[8] This distinction underscores that uncertainty about parameters arises from incomplete sampling, not from a probabilistic distribution over \theta itself.^[6] The framework delineates aleatory uncertainty, which stems from inherent randomness in the sampling process and is quantified via probabilities of observable outcomes, from epistemic uncertainty, which reflects ignorance about the fixed parameter value and is addressed through procedures guaranteeing performance in repeated trials. Aleatory variability captures the irreducible noise in data generation, while epistemic aspects are handled indirectly by ensuring methods control error rates over long runs, without modeling parameter uncertainty probabilistically.^[29] Central to this paradigm is the behavioristic interpretation, as articulated by Jerzy Neyman, which views statistical procedures as rules for inductive behavior that assure long-run coverage properties, such as confidence intervals enclosing the true parameter with a predetermined frequency across repetitions. These procedures prioritize the objective guarantee of error control in hypothetical ensembles, guiding actions like decision-making in scientific inquiry based on the anticipated performance of the method rather than epistemic probabilities for individual cases.^[30]

Interpretations and Debates

One of the central divides within frequentist inference concerns the approaches of Ronald A. Fisher and the Neyman-Pearson framework, particularly in their contrasting views on inductive inference versus inductive behavior. Fisher emphasized inductive inference through significance testing, using p-values to quantify evidence against a null hypothesis and aiming to draw conclusions about specific hypotheses based on evidential strength, as articulated in his 1935 work where he described tests as tools for "inductive reasoning" to infer the truth or falsehood of propositions. In contrast, the Neyman-Pearson approach focused on inductive behavior, prioritizing long-run error control (Type I and Type II errors) via decision rules that ensure reliable performance across repeated applications, without claiming probabilistic statements about particular parameters or hypotheses. This distinction led to ongoing tensions, with Fisher criticizing Neyman-Pearson methods for reducing inference to mechanical rule-following that ignores evidential context, while Neyman viewed Fisher's approach as overly subjective and prone to fiducial inconsistencies.^[31] A related debate centers on the interpretation of confidence intervals, pitting the strict adherence to coverage probability against any prohibition on assigning degrees of belief to the interval for a given dataset. In the frequentist paradigm, a 95% confidence interval is interpreted solely in terms of long-run frequency: the method that generates it will contain the true parameter in 95% of repeated samples from the same population, as formalized by Neyman in 1937. This view explicitly prohibits interpreting the observed interval as having a 95% probability of containing the true value post-data, deeming such statements as a "fundamental confidence fallacy" because the interval is fixed while the parameter is unknown, rendering the probability either 0 or 1.^[32] Defenders of this interpretation argue it maintains objectivity by avoiding subjective probabilities, yet critics within frequentism note that this restriction can hinder practical communication, leading to calls for more nuanced evidential readings without crossing into Bayesian territory. Fisher's fiducial argument, introduced in 1930 as a method to invert probability statements from data to parameters without priors, faced substantial critiques that contributed to its partial abandonment after the 1950s. The argument posited that certain pivotal quantities allow direct fiducial distributions for parameters, treating them as if they had objective probabilities derived from the sampling distribution.^[33] However, extensions to multiparameter cases revealed paradoxes, such as non-uniqueness of fiducial distributions and conflicts with conditioning principles, as highlighted by Bartlett in 1936 and further exposed in Stein's 1959 critique of the Behrens-Fisher problem.^[25] By the late 1950s, these issues, compounded by the Buehler-Feddersen 1963 disproof of Fisher's "recognizable subsets" justification, led to widespread rejection among frequentists, who favored confidence intervals as a more robust alternative despite shared foundational challenges.^[25] Modern developments, such as generalized fiducial inference since the early 2000s, have sought to revive and formalize these ideas to resolve classical paradoxes while preserving frequentist principles.^[34] In modern frequentist testing, a persistent debate revolves around conditional versus unconditional error rates, reflecting tensions over the relevance of error control to specific data versus overall procedures. Unconditional error rates, as in the Neyman-Pearson framework, average Type I errors across all possible ancillary statistics or experimental frames, providing global guarantees but potentially diluting relevance to the observed data.^[28] Conditional error rates, advocated by proponents like Birnbaum and Cox, condition on observed ancillaries to ensure error probabilities reflect the specific experimental context, aligning inference more closely with the likelihood principle and avoiding misleading inferences from irrelevant averaging.^[35] This debate underscores unresolved foundational issues, with conditional approaches gaining traction in complex models for their informativeness, though unconditional methods remain standard for their simplicity and long-run validity.^[28] A pivotal event in these intra-frequentist debates was Leonard J. Savage's 1962 critique in "The Foundations of Statistical Inference," which exposed foundational fragilities and elicited defensive responses from the community. Savage argued that frequentist methods suffer from disunity—evident in the Fisher-Neyman schism—and fail to resolve subjective elements like choice of test or stopping rules, rendering concepts like confidence levels practically empty without personal probabilities.^[36] He illustrated this with examples where mechanical confidence intervals yield counterintuitive results, such as overly wide credible bounds from minimal data, and advocated Bayesian unification over fragmented frequentist tools.^[36] Responses from figures like E.S. Pearson and G.A. Barnard defended frequentism's objective frequencies and developmental potential, acknowledging flaws but emphasizing its utility in empirical sciences, which spurred refinements in error control and conditioning principles throughout the 1960s and beyond.^[36]

Inference Methods

Hypothesis Testing Frameworks

In frequentist hypothesis testing, the goal is to decide between a null hypothesis H_0: \theta \in \Theta_0 and an alternative hypothesis H_1: \theta \in \Theta_1, where \theta represents the unknown parameter of interest and \Theta_0, \Theta_1 are disjoint subsets of the parameter space.^[24] This framework treats the hypotheses as fixed statements about the population, with decisions based on observed data from a random sample. The procedure controls the risk of incorrect decisions through predefined error rates, emphasizing long-run frequency properties over the specific data realization. The framework defines two types of errors: Type I error, which occurs when H_0 is rejected despite being true, and Type II error, when H_0 is not rejected despite H_1 being true.^[24] The significance level \alpha is the probability of a Type I error, formally \alpha = P(\text{reject } H_0 \mid H_0 \text{ true}), typically set to a small value like 0.05 to limit false positives. The power of the test, $1 - \beta = P(\text{reject } H_0 \mid H_1 \text{ true}), measures the probability of correctly detecting the alternative, where \beta is the Type II error rate; higher power is desirable but often trades off against \alpha.^[24] A test statistic T is computed from the data, and rejection of H_0 occurs if T falls into a critical region determined by \alpha. The p-value provides a measure of evidence against H_0, defined as p = P(T \geq t_{\text{obs}} \mid H_0), where t_{\text{obs}} is the observed value of the test statistic; small p-values (e.g., below \alpha) suggest rejecting H_0. This approach, rooted in Ronald Fisher's work, quantifies the extremeness of the data under the null without fixing \alpha in advance. The Neyman-Pearson lemma provides a foundation for optimal tests, stating that for simple hypotheses (specific points in \Theta_0 and \Theta_1), the likelihood ratio test rejects H_0 when \frac{L(\theta_1 | \mathbf{x})}{L(\theta_0 | \mathbf{x})} > k, where L is the likelihood function and k is chosen to achieve size \alpha; this yields the uniformly most powerful (UMP) test among those of size \alpha.^[24] For one-sided alternatives in exponential families, UMP tests exist and extend this principle, maximizing power while controlling \alpha. However, UMP tests are not always available for composite hypotheses, leading to alternative criteria like unbiasedness. A classic example is the one-sample t-test for testing H_0: \mu = \mu_0 against H_1: \mu > \mu_0, where the test statistic is

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}},

with \bar{x} the sample mean, s the sample standard deviation, and n the sample size; under H_0, t follows a Student's t-distribution with n-1 degrees of freedom.^[37] Rejection occurs if t > t_{\alpha, n-1}, the critical value from the t-table. When multiple hypotheses are tested simultaneously, the family-wise error rate can inflate beyond \alpha, necessitating adjustments. The Bonferroni correction addresses this by dividing \alpha by the number of tests (e.g., \alpha/m for m tests), conservatively controlling the probability of any Type I error across the family while reducing power for individual tests. This method, derived from probability inequalities, provides a simple conceptual tool for multiple comparisons but is often critiqued for its stringency in large-scale testing.

Confidence Intervals and Estimation

In frequentist inference, point estimation seeks to approximate an unknown parameter θ using a statistic θ̂ derived from the observed data X. An estimator is unbiased if its expected value equals the true parameter, E[θ̂] = θ, ensuring that, on average over repeated samples, the estimate centers on the truth. This property was emphasized in early foundational work on statistical estimation. Consistency provides a stronger guarantee for large samples, requiring that θ̂ converges in probability to θ as the sample size n increases, denoted plim_{n→∞} θ̂ = θ; this criterion ensures the estimator becomes arbitrarily reliable with more data. Ronald Fisher introduced the concept of consistency in his 1922 paper, highlighting its role in validating estimators like the sample mean for the population mean under suitable conditions. Interval estimation extends point estimation by providing a range of plausible values for θ, accounting for sampling variability through confidence intervals. A (1 - α)100% confidence interval CI(X) is constructed such that, in repeated sampling from the fixed population, the true θ lies within CI(X) with probability 1 - α: P(θ ∈ CI(X)) = 1 - α. This frequentist coverage probability emphasizes long-run performance rather than a probability statement about the specific interval observed. Jerzy Neyman formalized this approach in 1937, defining confidence intervals as procedures with guaranteed coverage across hypothetical repetitions. Confidence intervals can be derived by inverting hypothesis tests, where the interval comprises all θ₀ values for which the null hypothesis H₀: θ = θ₀ is not rejected at significance level α using a suitable test statistic. This duality links estimation directly to testing frameworks, ensuring the interval aligns with the tests' error control properties. Neyman's theory integrated this inversion principle to yield intervals with optimal coverage. For instance, when estimating the mean μ of a normal distribution with unknown variance σ² based on a sample of size n, the (1 - α)100% confidence interval is \bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}, where \bar{x} is the sample mean, s is the sample standard deviation, and t_{\alpha/2, n-1} is the critical value from the t-distribution with n-1 degrees of freedom. This interval, rooted in William Sealy Gosset's 1908 derivation of the t-distribution, achieves exact coverage under normality assumptions. Assessing point estimators often involves the bias-variance tradeoff, captured by the mean squared error MSE(θ̂) = Var(θ̂) + [Bias(θ̂)]², which quantifies total estimation error as the sum of variability and systematic deviation from θ. Reducing bias may increase variance, necessitating choices that minimize MSE for finite samples; this decomposition guides estimator selection in frequentist practice. For large n, many estimators exhibit asymptotic normality via the central limit theorem, where \sqrt{n} (θ̂ - θ) \xrightarrow{d} N(0, V) for some variance V, enabling approximate confidence intervals like θ̂ \pm z_{\alpha/2} \sqrt{\widehat{V}/n} using the standard normal quantile z_{\alpha/2}. This property underpins much of modern frequentist inference, as articulated in early asymptotic theory.

Sufficient Statistics and Likelihood

In frequentist inference, a sufficient statistic T(\mathbf{X}) for a parameter \theta based on observed data \mathbf{X} is defined as a function of the data such that the conditional distribution of \mathbf{X} given T(\mathbf{X}) = t is independent of \theta. This property implies that T(\mathbf{X}) captures all the information about \theta contained in \mathbf{X}, allowing for data reduction without loss of inferential value. The concept was introduced by Ronald A. Fisher to facilitate efficient estimation by focusing on reduced-dimensional summaries of the data. The Fisher-Neyman factorization theorem provides a practical criterion for identifying sufficient statistics. It states that a statistic T(\mathbf{X}) is sufficient for \theta if and only if the joint probability density (or mass) function of \mathbf{X} can be factored as

f(\mathbf{x} \mid \theta) = h(\mathbf{x}) \cdot g(\theta, T(\mathbf{x})),

where h(\mathbf{x}) does not depend on \theta, and g is a function involving both \theta and T(\mathbf{x}). Fisher originally derived this for specific cases in likelihood-based estimation, while Jerzy Neyman extended it to more general settings, establishing its broad applicability in verifying sufficiency. The maximum likelihood estimator (MLE) arises naturally in the context of sufficient statistics and the likelihood function. The likelihood L(\theta; \mathbf{X}) = f(\mathbf{X} \mid \theta) measures how well a parameter value explains the observed data, and the MLE \hat{\theta} is defined as

\hat{\theta} = \arg\max_{\theta} L(\theta; \mathbf{X}).

Fisher introduced the MLE as an efficient method for point estimation, noting its desirable invariance properties: if \hat{\theta} is the MLE of \theta, then for any function r(\cdot), r(\hat{\theta}) is the MLE of r(\theta). This invariance ensures consistency under reparametrization, making the MLE a cornerstone of frequentist estimation. Under regularity conditions, the MLE demonstrates asymptotic efficiency, achieving the Cramér-Rao lower bound (CRLB) for the variance of unbiased estimators. The CRLB states that for an unbiased estimator \hat{\theta} of \theta, its variance satisfies

\text{Var}(\hat{\theta}) \geq \frac{1}{I(\theta)},

where I(\theta) = \mathbb{E}\left[ -\frac{\partial^2}{\partial \theta^2} \log L(\theta; \mathbf{X}) \right] is the Fisher information, quantifying the amount of information about \theta in the data. The MLE \hat{\theta} is asymptotically normally distributed with mean \theta and variance $1/(n I(\theta)) for sample size n, saturating the bound and thus attaining minimal asymptotic variance. This bound was independently derived by C. R. Rao and Harald Cramér, highlighting the efficiency limit for frequentist estimators. Sufficient statistics are particularly tractable in the exponential family of distributions, where the probability density takes the form

f(\mathbf{x} \mid \theta) = h(\mathbf{x}) \exp\left( \eta(\theta) T(\mathbf{x}) - A(\theta) \right),

directly satisfying the factorization theorem with T(\mathbf{x}) as sufficient. For the normal distribution N(\mu, \sigma^2) with known variance, the sample mean \bar{X} is sufficient for \mu, as the likelihood factors through \sum X_i. Similarly, for the binomial distribution \text{Bin}(n, p), the number of successes S = \sum X_i is sufficient for p, reducing the data to a single scalar while preserving all information about the success probability. These examples illustrate how exponential family structure enables explicit identification of low-dimensional sufficient statistics, enhancing computational efficiency in inference. The Rao-Blackwell theorem further leverages sufficient statistics to improve estimators. It asserts that if \delta(\mathbf{X}) is an unbiased estimator of \theta and T(\mathbf{X}) is sufficient, then the conditional expectation \delta^*(\mathbf{X}) = \mathbb{E}[\delta(\mathbf{X}) \mid T(\mathbf{X})] is also unbiased but has variance no larger than that of \delta(\mathbf{X}), with equality only if \delta is already a function of T. This theorem, developed by C. R. Rao and David Blackwell, provides a method to "Rao-Blackwellize" crude estimators, yielding more efficient alternatives by conditioning on the sufficient statistic, thereby reducing mean squared error without introducing bias.

Applications and Design

Experimental Methodology

In frequentist experimental methodology, randomization serves as a foundational principle to ensure unbiased allocation of treatments and control for extraneous variability, thereby enabling valid inference about treatment effects. By randomly assigning experimental units to treatment groups, researchers mitigate selection bias and allow the use of randomization-based tests to assess significance under the null hypothesis of no treatment effect.^[38] This approach, pioneered by Ronald Fisher, underpins the design of experiments where systematic assignment could otherwise confound results.^[39] To further control variability, blocking groups experimental units into homogeneous subsets based on known sources of variation, such as soil type in agricultural trials, ensuring that treatment effects are estimated more precisely within each block.^[40] Factorial designs extend this by simultaneously varying multiple factors at different levels, allowing estimation of main effects and interactions while maximizing efficiency in resource use; for instance, a 2x2 factorial examines two binary factors across all combinations.^[41] These techniques, integral to Fisher's framework, facilitate the partitioning of total variance into components attributable to treatments, blocks, and residual error.^[40] Power analysis is employed to determine the appropriate sample size n prior to the experiment, ensuring sufficient power $1 - \beta to detect a specified effect size at a chosen significance level \alpha. In the Neyman-Pearson framework, this involves balancing the risks of Type I and Type II errors, where larger n reduces \beta for a fixed effect size, thus guiding resource allocation to achieve reliable detection of meaningful differences. Such pre-experiment planning upholds frequentist validity by quantifying the experiment's sensitivity to true effects. Replication is emphasized in frequentist designs as each experiment represents one realization in an infinite series of identical replications under the same conditions, with long-run frequencies validating the procedure's error rates. This perspective, contrasting with one-off analyses, underscores the need for multiple observations per treatment to estimate variance and achieve stable frequency-based inferences, as articulated in Neyman's behavioral interpretation of tests. To control confounding variables, randomization tests evaluate the observed data against all possible outcomes under random assignment, providing an exact distribution-free assessment of significance. Fisher's exact test exemplifies this for categorical data in contingency tables, computing the probability of the observed table or more extreme under the null, thereby isolating treatment effects without parametric assumptions.^[38] This method ensures that any deviation from the null arises from treatment rather than systematic biases. For multi-factor experiments, analysis of variance (ANOVA) frameworks decompose total variability into additive components for factors, interactions, and error, using F-tests to assess significance while maintaining control over the experiment-wide Type I error rate. Fisher's development of ANOVA accommodates complex designs, such as randomized block or factorial layouts, by modeling variance partitions that support inference on multiple effects simultaneously.^[40] In modern contexts, adaptive designs incorporate pre-specified stopping rules to modify trial parameters, such as sample size or arms, based on interim data while preserving frequentist control of error rates through methods like alpha-spending functions.^[42] These designs, guided by regulatory frameworks, allow flexibility in clinical trials—e.g., early termination for efficacy or futility—provided adaptations are prospectively defined to avoid inflated Type I errors.^[42]

Practical Examples

One illustrative example of frequentist inference is the interpretation of p-values through the long-run frequency property, demonstrated using a sequence of fair coin flips. Suppose a researcher tests the null hypothesis that a coin is fair (probability of heads p = 0.5) by flipping it repeatedly and computing the p-value for observing 16 or more heads in 20 flips, which is approximately 0.010 under the null. In repeated simulations of this experiment under the true null, the p-value will fall below 0.05 about 5% of the time, reflecting the long-run relative frequency of Type I errors across hypothetical replications.^[43] In clinical trials, frequentist methods like the t-test and confidence intervals are commonly applied to assess drug efficacy by comparing mean outcomes between treatment and control groups. For instance, consider a randomized controlled trial evaluating a new analgesic drug's effect on pain reduction scores, where the null hypothesis states no difference in mean scores between the drug and placebo groups. Researchers perform an independent samples t-test on data from approximately 100 participants per group, yielding a statistically significant result (p < 0.05) that rejects the null, indicating the drug reduces pain. A 95% confidence interval for the mean difference provides a range of plausible values for the true effect in the population.^[44] A/B testing in technology companies exemplifies the use of chi-square tests for categorical outcomes, such as conversion rates on websites. In a typical setup, Version A (control) is shown to 10,000 users with a 5% conversion rate (500 conversions), while Version B (treatment) is shown to another 10,000 users with a 5.2% rate (520 conversions). The chi-square test assesses the null hypothesis of no association between version and conversion, producing a statistic of approximately 4.0 (df = 1, p ≈ 0.046), which rejects the null at the 0.05 level and supports Version B's superiority. This p-value interpretation guides decisions on deployment, emphasizing the method's role in controlling false positives over many such tests.^[45] In economics, ordinary least squares (OLS) regression with the F-test evaluates model fit for relationships like wages and education. A seminal application involves regressing log wages on years of schooling, experience, and tenure using panel data from the National Longitudinal Survey of Youth. The OLS estimates provide coefficients (e.g., 0.08 for schooling, indicating an 8% wage increase per year), and the overall F-test (F = 45.2, df = 3 and 2,365, p < 0.001) rejects the null of no explanatory power, confirming the model's significant fit to the data and enabling inference on economic returns to human capital.^[46] Genome-wide association studies (GWAS) apply frequentist inference through multiple testing corrections to identify genetic variants linked to traits, using the false discovery rate (FDR) procedure. In a study scanning approximately 592,000 single nucleotide polymorphisms (SNPs) for associations with type 2 diabetes using UK Biobank data, raw p-values were adjusted to control the FDR, identifying hundreds of discoveries (e.g., 940 at 10% FDR) while balancing discovery power against multiplicity.^[47] A recent application in the 2020s involves frequentist analysis in COVID-19 vaccine trials, where confidence intervals quantify efficacy rates. In the Pfizer-BioNTech phase 3 trial with over 44,000 participants, the vaccine group had 8 infections versus 162 in the placebo group, yielding a vaccine efficacy estimate of 95% with a 95% confidence interval of [90.3%, 97.6%]. This interval, derived from the Clopper-Pearson method, supports regulatory approval by excluding lower bounds below 50% efficacy, demonstrating the approach's role in providing frequentist guarantees for public health decisions.

Comparisons and Critiques

Relation to Bayesian Inference

Frequentist inference treats parameters as fixed but unknown quantities, deriving inferences based solely on the likelihood of observed data under repeated sampling, whereas Bayesian inference views parameters as random variables governed by a prior distribution \pi(\theta) that is updated with the data via the likelihood L(\theta; X) to yield a posterior distribution P(\theta | X) \propto L(\theta; X) \pi(\theta). This fundamental difference leads to Bayesian methods incorporating subjective or objective prior beliefs about parameters before observing data, a practice rejected by frequentists who argue that priors introduce unverifiable assumptions not grounded in the data alone. In decision-theoretic terms, Bayesian approaches emphasize pre-posterior analysis, optimizing expected loss over the posterior distribution to guide actions like hypothesis selection, while frequentist methods focus on long-run error rates, such as Type I and Type II errors, across hypothetical repeated experiments to ensure procedures like tests and intervals have controlled frequentist coverage properties. This contrast often results in differing conclusions; for instance, in estimating a binomial proportion p from n trials with k successes, a frequentist 95% confidence interval might use the Clopper-Pearson method to provide an interval that covers the true p in 95% of repeated samples, whereas a Bayesian analysis with a Beta($1,1) uniform prior yields a posterior Beta(k+1, n-k+1) credible interval that directly quantifies updated belief about p, potentially narrower or shifted depending on the prior. A key point of convergence is the correspondence principle, where Bayesian procedures using non-informative or reference priors—designed to be minimally influential—often approximate frequentist results, particularly in large samples where the likelihood dominates the posterior.^[48] However, historical tensions highlight divergences, as exemplified by Lindley's paradox, where for testing a point null hypothesis against a composite alternative with large sample sizes, a frequentist test may reject the null at a 5% significance level due to a small p-value, yet the corresponding Bayesian analysis with a broad prior favors the null by assigning it higher posterior odds, illustrating how priors can override data-driven evidence in high-information scenarios.^[49] Modern developments include empirical Bayes methods, which estimate priors from the data itself to bridge the paradigms, treating hyperparameters as fixed in a frequentist manner while performing Bayesian updates on parameters, though frequentists critique this as still introducing subjectivity through data-dependent prior selection without guaranteed long-run properties.

Criticisms and Alternatives

One major criticism of frequentist inference centers on the frequent misinterpretation of p-values as posterior probabilities or direct measures of the probability that the null hypothesis is true. In reality, a p-value represents the probability of observing data as extreme as or more extreme than the sample data, assuming the null hypothesis is true, but it does not quantify the probability that the null hypothesis is correct or the strength of evidence against it in a posterior sense. This confusion has led to widespread overinterpretation, where small p-values are taken as strong evidence for an alternative hypothesis, contributing to erroneous conclusions in scientific literature. The American Statistical Association's 2016 statement explicitly warns against such misuses, emphasizing that p-values alone do not measure effect size or the probability of a hypothesis being true. A related critique applies to confidence intervals in frequentist statistics, where they are often misinterpreted as providing the probability that the true parameter lies within the interval for a specific realization of the data. Frequentist theory defines a confidence interval as the result of a procedure that covers the true parameter with the stated probability (e.g., 95%) over repeated sampling, but for any single computed interval, the true parameter either is or is not contained within it—there is no probabilistic coverage guarantee for that particular instance. This disconnect between the long-run frequency interpretation and intuitive expectations about individual intervals has been highlighted as a fundamental flaw, leading researchers to assign undue certainty to observed intervals. Frequentist methods are also vulnerable to optional stopping and p-hacking, practices where researchers adjust data collection or analysis flexibly to achieve statistical significance without pre-specifying protocols. Optional stopping involves continuing data collection until a p-value drops below a threshold, inflating the Type I error rate beyond nominal levels, while p-hacking includes selective reporting of analyses or outcomes that yield significant results. These behaviors exploit the flexibility in frequentist hypothesis testing, undermining the validity of inferences, as demonstrated in simulations showing that common questionable research practices can produce false positives in over 60% of studies. As an alternative to frequentism, the likelihoodist approach, proposed by Birnbaum in 1962, emphasizes relative evidential support through likelihood ratios without relying on long-run frequencies or hypothetical repetitions. Under this framework, inference is based solely on how well observed data support different parameter values via the likelihood function, adhering to the likelihood principle that experimental conclusions should depend only on the likelihoods of the observed data. This avoids the sampling distribution dependencies of frequentism, focusing instead on direct comparisons of support for competing hypotheses. In response to these criticisms, particularly amid the 2010s replication crisis in psychology where only about 36% of studies replicated significant effects, frequentists have advanced pre-registration and reproducibility initiatives to mitigate p-hacking and optional stopping. Pre-registration requires researchers to specify hypotheses, sample sizes, and analysis plans in advance on public platforms like the Open Science Framework, reducing flexibility and enhancing transparency, as evidenced by improved replication rates in preregistered studies. These reforms aim to preserve the strengths of frequentist error control while addressing practical abuses. A modern development reflecting this shift is the American Psychological Association's Journal Article Reporting Standards for quantitative research (JARS-Quant, 2018) and the 7th edition Publication Manual (2020), which prioritize estimation-based reporting—such as effect sizes and confidence intervals—over exclusive reliance on null hypothesis significance testing (NHST). This guidance encourages comprehensive presentation of uncertainty and practical significance, aligning with calls to move beyond dichotomous p-value decisions to foster more robust inference.^[50]

References

[1]
[PDF] Bayesian Versus Frequentist Inference - Eric-Jan Wagenmakers
Frequentist inference is based on the idea that probability is a limiting fre- quency. This means that a frequentist feels comfortable assigning probability to ...Missing: scholarly | Show results with:scholarly<|control11|><|separator|>
[2]
Frequentist statistical inference without repeated sampling | Synthese
Mar 11, 2022 · Frequentist inference typically is described in terms of hypothetical repeated sampling but there are advantages to an interpretation that uses a single random ...
[3]
Understanding the Differences Between Bayesian and Frequentist ...
Frequentist inference begins by assuming a null hypothesis to be true before data are collected (eg, that there is no effect of a particular treatment on ...
[4]
Frequentist statistics as a theory of inductive inference - Project Euclid
By contrast, a central feature of frequentist statistics is to be able to assess and control the probability that a test would have rejected a hypothesis, if ...Missing: scholarly | Show results with:scholarly
[5]
[PDF] Statistical Methods For Research Workers Thirteenth Edition
Page 1. Statistical Methods for. Research Workers. BY. Sir RONALD A. FISHER, sg.d., f.r.s.. D.Sc. (Ames, Chicago, Harvard, London), LL.D. (Calcutta, Glasgow).
[6]
[PDF] Outline of a Theory of Statistical Estimation Based on the Classical ...
The theory of statistical estimation, based on classical probability, involves determining numerical values of parameters from experimental data, using a ...
[7]
Interpreting frequentist hypothesis tests: insights from Bayesian ...
Oct 4, 2023 · Statistical inference is the process of analyzing samples to infer characteristics about the populations from which the samples are drawn.
[8]
[PDF] Frequentist Probability and Frequentist Statistics
Jul 9, 2024 · [17] Joint Statistical Papers of J. Neyman and E. S. Pearson, University of California Press,. Berkeley, 1967. [18] Neyman, J., ...
[9]
[PDF] Statistical Inference
Chapters 7-9 represent the central core of statistical inference, estimation (point and interval) and hypothesis testing. A major feature of these chapters ...Missing: frequentist | Show results with:frequentist
[10]
Long-Run Frequency - an overview | ScienceDirect Topics
The frequentist interpretation identifies the probability of an event A with the limit of the relative frequency of its occurrence: (14) P ( A ) : = lim n ...
[11]
[PDF] Frequentist statistics: a concise introduction - UBC Computer Science
Sep 5, 2007 · . Later we will see that many sampling distributions are approximately Gaussian as the sample size goes to infinity. More precisely, we say ...
[12]
VON MISES' AXIOMATISATION OF RANDOM SEQUENCES
We discuss von Mises' notion of a random sequence in the context of his approach to probability theory. We claim that the acceptance.
[13]
[PDF] Von Mises' Frequentist Approach to Probability - StatLit.org
The word 'chance' is used by von Mises to describe any process having a 'stable limiting frequency.' First basic assumption. Consider a finite label space S ...
[14]
[PDF] 2 Roots of Randomness: Von Mises' Definition of Random Sequences
In particular, perhaps somewhat surprisingly, the term "probability" does not occur in von Mises' axioms, but is a defined notion, whereas it is a primitive ...<|control11|><|separator|>
[15]
Special Distributions | Bernoulli Distribution | Binomial Distribution
Suppose that I have a coin with P(H)=p. I toss the coin n times and define X to be the total number of heads that I observe. Then X is binomial with parameter n ...
[16]
[PDF] A Tricentenary history of the Law of Large Numbers - arXiv
The Weak Law of Large Numbers is traced chronologically from its inception as Jacob Bernoulli's. Theorem in 1713, through De Moivre's Theorem, ...
[17]
[PDF] The Early Development of Mathematical Probability - Glenn Shafer
Bernoulli advanced this theorem (later called the law of large numbers by Poisson) as a justification for using observed frequencies as probabilities, to be ...
[18]
[PDF] 1 S.-D. Poisson Researches into the Probabilities of Judgements in ...
3. The law of large numbers and the central limit theorem. For. Poisson, that law was rather a loose principle, and for many decades.<|separator|>
[19]
The Mathematics of Society: Variation and Error in Quetelet's Statistics
probability theory, and hence the fundamental axiom of social physics, was the law of large numbers: over the long run, the frequency of events of any given ...<|separator|>
[20]
[PDF] Laplace: direct and inverse probabilities
The left side probability is the “inverse probability” with respect to the right side probability. Hans Fischer. Laplace: direct and inverse probabilities. März ...
[21]
Gauss's Derivation of the Normal Distribution and the Method of ...
It has been suggested that Gauss used the method of least squares on a data set published in 1799. The data set and its adjustment are reexamined, and it is ...Abstract · References (0) · The Method Of Gauss In 1799
[22]
On the mathematical foundations of theoretical statistics - Journals
A recent paper entitled "The Fundamental Problem of Practical Statistics," in which one of the most eminent of modern statisticians presents what purports to ...
[23]
Fisher (1925) Chapter 3 - Classics in the History of Psychology
Critical tests of this kind may be called tests of significance, and when such tests are available we may discover whether a second sample is or is not ...
[24]
IX. On the problem of the most efficient tests of statistical hypotheses
The problem of testing statistical hypotheses is an old one. Its origin is usually connected with the name of Thomas Bayes.
[25]
R. A. Fisher and Fiducial Argument - Project Euclid
The fiducial argument arose from Fisher's desire to create an inferential alternative to inverse methods. Fisher discovered such an alternative in 1930 ...
[26]
Karl Pearson and R. A. Fisher on Statistical Tests: A 1935 Exchange ...
In 1935, a letter to Nature criticizing the logic of statistical tests provoked published responses from Karl Pearson and RA Fisher.Missing: correspondence debates
[27]
Comments on the Neyman-Fisher Controversy and Its Consequences
The Neyman–Fisher controversy considered here originated with the 1935 presentation of Jerzy Neyman's Statistical Problems in Agricul- tural Experimentation to ...Missing: correspondence | Show results with:correspondence
[28]
[PDF] II Objectivity and Conditionality in Frequentist Inference
The priors are not to be considered expressions of uncertainty, ignorance, or degree of belief. Conventional priors may not even be probabilities in that a ...
[29]
Ale Epi - UC Berkeley Statistics
Epistemic refers to lack of knowledge -- something we could in principle know for sure -- in contrast to aleatoric intrinsic randomness involved in which of ...
[30]
[PDF] Frequentist Statistics as a Theory of Inductive Inference - Blogs at Kent
E. S. Pearson [23] is known to have disassociated himself from a narrow behaviourist interpretation (Mayo [15]). Neyman, at least in his discussion.
[31]
[PDF] Fisher, Neyman and Pearson - Error Statistics Philosophy
Nov 18, 2019 · (*If statistical inference is Bayesian, Neyman will talk instead of inductive "behavior"). 2. To avoid the pitfalls of Fisher's fiducial ...
[32]
The fallacy of placing confidence in confidence intervals - PMC
Frequentist CI theory says nothing at all about the probability that a particular, observed confidence interval contains the true value; it is either 0 (if the ...
[33]
Fiducial theory and optimal inference - Project Euclid
Fiducial theory was introduced by Fisher (1930) to avoid the problems related to the choice of a prior distribution. Fiducial inference has not gained much ...Missing: critiques post-
[34]
Could Fisher, Jeffreys and Neyman Have Agreed on Testing?
The resulting conditional frequentist error probabilities equal the objective posterior probabilities of the hypotheses advocated by Jeffreys. Key words and ...Missing: modern | Show results with:modern
[35]
[PDF] The Foundations of Statistical Inference
This monograph, based on a lecture, discusses subjective probability and statistical inference, which is how we find things out. It has three parts.
[36]
THE PROBABLE ERROR OF A MEAN | Biometrika - Oxford Academic
STUDENT; THE PROBABLE ERROR OF A MEAN, Biometrika, Volume 6, Issue 1, 1 March 1908, Pages 1–25, https://doi.org/10.1093/biomet/6.1.1.
[37]
[PDF] Causal Inference Chapter 2.1. Randomized Experiments: Fisher's ...
▷ RA Fisher was the first to grasp the importance of randomization for credibly assessing causal effects (1925, 1935). ▷ Given data from such a randomized ...
[38]
R. A. Fisher and Experimental Design: A Review - jstor
R. A. Fisher's contributions to experimental design are surveyed, particular attention being paid to. (1) the basic principles of replication, ...
[39]
[PDF] The Design of Experiments By Sir Ronald A. Fisher.djvu
Statistical procedure and experimental design are only two different aspects of the same whole, and that whole comprises all the logical requirements of the ...Missing: blocking | Show results with:blocking
[40]
R.A. Fisher and the Design of Experiments, 1922–1926
Mar 26, 2012 · The article indicates the radically new form and efficiency of factorial block designs, shows the further advantages accruing to factorial ...
[41]
[PDF] Adaptive Designs for Clinical Trials of Drugs and Biologics - FDA
As with other adaptive designs, the adaptation rule should be prespecified, and statistical hypothesis testing methods should account for the adaptive ...
[42]
Détente: A Practical Understanding of P values and Bayesian ... - NIH
Sep 26, 2020 · The P value is calculated as pr(n consecutive H's | H 0 is true) = 0.5n. b. The pr(biased coin) is calculated using Bayes formula (see ...Missing: run | Show results with:run<|separator|>
[43]
Sample Size Estimation in Clinical Trial - PMC - NIH
Let us see an example. Dr. ABC had developed a drug NEW which was effective in reducing pain. The drug NEW is clinically better in terms of efficacy and safety ...
[44]
Approaches to analyzing binary data for large-scale A/B testing - NIH
Abstract. An industry-academic collaboration was established to evaluate the choice of statistical test and study design for A/B testing in larger-scale ...
[45]
[PDF] Analysis of Economics Data: An Introduction to Econometrics
... tests that are used to extrapolate from the sample at hand to the population. The fourth part of the book presents material beyond basic regression. The ...
[46]
False discovery rate control in genome-wide association ... - PNAS
The naive approach is to apply the Benjamini–Hochberg (BH) correction (19) to the marginal P values before clumping. We shall see this inflates the type-I ...
[47]
[PDF] Default priors for Bayesian and frequentist inference
Summary. We investigate the choice of default priors for use with likelihood for Bayesian and frequentist inference. Such a prior is a density or relative ...
[48]
A Statistical Paradox - jstor
This is a journal article titled 'A Statistical Paradox' by D.V. Lindley, published in Biometrika in 1957.