Abstract
Statistical modeling is often used to measure the strength of evidence for or against hypotheses about given data. We have previously proposed an information-dynamic framework in support of a properly calibrated measurement scale for statistical evidence, borrowing some mathematics from thermodynamics, and showing how an evidential analogue of the ideal gas equation of state could be used to measure evidence for a one-sided binomial hypothesis comparison (“coin is fair” vs. “coin is biased towards heads”). Here we take three important steps forward in generalizing the framework beyond this simple example, albeit still in the context of the binomial model. We: (1) extend the scope of application to other forms of hypothesis comparison; (2) show that doing so requires only the original ideal gas equation plus one simple extension, which has the form of the Van der Waals equation; (3) begin to develop the principles required to resolve a key constant, which enables us to calibrate the measurement scale across applications, and which we find to be related to the familiar statistical concept of degrees of freedom. This paper thus moves our information-dynamic theory substantially closer to the goal of producing a practical, properly calibrated measure of statistical evidence for use in general applications.
Highlights
Statistical modeling is used for a variety of purposes throughout the biological and social sciences, including hypothesis testing and parameter estimation, among other things, but there is a distinct purpose to statistical inference, namely, measurement of the strength of evidence for or against hypotheses in view of data
With the results presented above, we have taken important steps forward towards generalizing our original information-dynamic theory in support of a properly calibrated measure E of statistical evidence
While the original EqS had the same form as the ideal gas equation, the revised EqS needed to properly handle nested hypothesis contrast (HC) has the same form as the thermodynamic Van der Waals equation
Summary
Statistical modeling is used for a variety of purposes throughout the biological and social sciences, including hypothesis testing and parameter estimation, among other things, but there is a distinct purpose to statistical inference, namely, measurement of the strength of evidence for or against hypotheses in view of data. (i) We have previously worked out a concrete application only for a simple coin-tossing model, and we speculated that extension to other statistical models (i.e., forms of the likelihood other than the binomial) might require derivation of a new underlying equation of state (EqS, that is, the formula for computing the evidence E; see below for details) for every distinct statistical model. In order to establish basic behavior patterns (BBPs) expected of any evidence measure, we consider a simple model and a series of thought experiments, or appeals to intuition. E is defined as the proportionality between (i) the change in a certain form of information with the influx of new data; and (ii) the entropy, such that the degree of E retains constant meaning across the measurement scale and, given the correct EqS, across applications.
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