Bernoulli’s golden theorem in retrospect: error probabilities and trustworthy evidence

Abstract

Bernoulli’s 1713 golden theorem is viewed retrospectively in the context of modern model-based frequentist inference that revolves around the concept of a prespecified statistical model ${\mathcal{M}}_{{{\varvec{\uptheta}}}} \left( {\mathbf{x}} \right)$, defining the inductive premises of inference. It is argued that several widely-accepted claims relating to the golden theorem and frequentist inference are either misleading or erroneous: (a) Bernoulli solved the problem of inference ‘from probability to frequency’, and thus (b) the golden theorem cannot justify an approximate Confidence Interval (CI) for the unknown parameter $\theta$, (c) Bernoulli identified the probability $P\left( A \right)$ with the relative frequency $\frac{1}{n}\sum\nolimits_{k = 1}^{n} {x_{k} }$ of event A as a result of conflating $f({\mathbf{x}}_{0} |\theta )$ with $f(\theta |{\mathbf{x}}_{0} ),$ where ${\mathbf{x}}_{0}$ denotes the observed data, and (d) the same ‘swindle’ is currently perpetrated by the p value testers. In interrogating the claims (a)–(d), the paper raises several foundational issues that are particularly relevant for statistical induction as it relates to the current discussions on the replication crises and the trustworthiness of empirical evidence, arguing that: [i] The alleged Bernoulli swindle is grounded in the unwarranted claim $\hat{\theta }_{n} \left( {{\mathbf{x}}_{0} } \right) \simeq \theta^{*} ,$ for a large enough n, where $\hat{\theta }_{n} \left( {\mathbf{X}} \right)$ is an optimal estimator of the true value $\theta^{*}$ of θ. [ii] Frequentist error probabilities are not conditional on hypotheses (H₀ and H₁) framed in terms of an unknown parameter θ since θ is neither a random variable nor an event. [iii] The direct versus inverse inference problem is a contrived and misplaced charge since neither conditional distribution $f({\mathbf{x}}_{0} |\theta )$ and $f(\theta |{\mathbf{x}}_{0} )$ exists (formally or logically) in model-based (${\mathcal{M}}_{{{\varvec{\uptheta}}}} \left( {\mathbf{x}} \right)$) frequentist inference.