In the world of data science, we often use complex models to make sense of the world. Inference is concerned with logical connections, which may or may not correspond to causal physical influences.
At the Ninth Colston Symposium, philosopher Karl Popper (1957) described his propensity interpretation of probability as purely objective, but avoided using the term physical influence. Instead, Popper argued that the probability of a particular face showing up when tossing a die is not a physical property of the die itself—as emphasized by Cramér (1946)—but rather an objective property of the entire experimental setup, that is, the die together with the method of tossing.
Furthermore, a well-known exposition of statistical mechanics (Penrose, 1979) adopts as a fundamental axiom that probabilities referring to the present time can depend only on what happened earlier, never on what happens later. Penrose considers this to be a necessary physical requirement for causality.
This highlights a profound debate about the very nature of probability. Popper argued that propensity—the likelihood of an event—is not just a matter of our incomplete knowledge (a subjective view) nor is it a simple property of an object—like the weight of a die. Instead, he saw it as a real, objective, physically real property of the entire system in which an event occurs. For Popper, the chance of rolling a six is a tangible feature of the die, the table, the air, and the act of tossing itself. He was searching for the physical, causal roots of probability.
In any event, logical inference that we advocate here is fundamentally different, both in perspective and in its conclusions, from the theory of physical causation put forward by Penrose and Popper. Clearly, logical inference is applicable in many situations where assuming physical causation would not be appropriate.
A key philosophical expectation in the logic of inference is that learning from shared evidence should bring rational observers’ beliefs closer together, even if they began with different priors. Formally, if two agents A and B hold different prior information $ I_A $ and $ I_B $, then for any new data $ D $, we might expect:
$$ \left| P(S|D I_A) - P(S|D I_B) \right| < \left| P(S|I_A) - P(S|I_B) \right| $$
where $ S $ is an event or hypothesis of interest.
Although this convergence can be verified in special cases, it is not guaranteed in general—the updating process may still maintain or even sometimes increase initial disagreement, depending on the learning process. Again, the problem is choice.
If someone chooses to ignore evidence, learning cannot occur—unlike in the propensity framework, where subjectivity is irrelevant.
If such a notion—that some underlying propensity exists—is made into a precise hypothesis, then our approach to probability theory can be used to analyze what follows from it, as E.T. Jaynes explains:
In all scientific disciplines, logical inference tends to have the broader range of application. While we acknowledge that physical effects can propagate only forward in time, logical inference works equally well whether applied to the past or the future.
Understanding the distinction is crucial. Choosing the wrong approach can lead to flawed strategies, wasted resources, and incorrect conclusions. Think of it like using a microscope when you need a telescope—both are powerful tools, but for entirely different jobs.
We advocate for the logic of consistent reasoning from incomplete information rather than ontological randomness.
In other words, we follow The Science of Logic as envisioned by E.T. Jaynes. Even though this approach does not contradict Kolmogorov’s principles, it seeks to establish a deeper logical foundation that allows for assigning probabilities through the analysis of incomplete information—a concept absent from Kolmogorov’s system. This foundation permits the theory to be extended in the ways required for modern applications.
This approach is at the very core of Actuarial Mathematics, most notably exemplified by credibility theory.
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