Mindscape 315 | Branden Fitelson on the Logic and Use of Probability

Episode Overview

• Science operates on inductive reasoning and probabilistic confirmation, not absolute proof, differentiating it from mathematics or logic.
• The discussion explores various notions of probability, emphasizing its role in assessing evidence and argument strength in science and philosophy.
• A core theme is the distinction between subjective prior beliefs and objective likelihood ratios (Bayes factors) as invariant measures of evidence strength.
• The podcast introduces a two-dimensional theory of argument strength, considering both the probability of a conclusion and the relevance of the evidence.
• It addresses classic philosophical challenges like Hume's problem of induction and Hempel's Raven Paradox, showing how quantitative Bayesian confirmation offers resolutions.

Key Concepts

• Science operates through inductive reasoning, inferring patterns from observations, and never "proves" things in the deductive sense of mathematics or logic. Scientific conclusions are always open to revision.
• There are multiple kinds of probabilities (e.g., in biology, physics, everyday reasoning), but probability can also be viewed as a theoretical quantity defined by its functional role within a theory.
• Evaluating competing theories of probability or scientific hypotheses requires a "neutral" or "epistemic" notion of probability to avoid question-begging.
• The frequency theory of probability, defining probability by the actual frequency of events, faces challenges with finite sequences and unique events, as frequencies are primarily how we know about probabilities, not what they fundamentally are.
• Confirmational power refers to how much new evidence strengthens belief in a hypothesis, with objective "likelihood ratios" or "Bayes factors" quantifying this strength independently of subjective prior beliefs.
• The "Base Rate Fallacy" is a cognitive bias where people often neglect prior probabilities; ironically, the "Bayes factor" itself derives from objective error rates and provides the invariant information science uses.
• Karl Popper's philosophy of falsification correctly recognizes the asymmetry and power of refuting evidence but erred in dismissing the existence and quantifiability of positive confirmation.
• The "dream" of a single, universal probability function to measure all argument strengths is considered absurd; a pluralist approach where suitable probability functions exist for each argument context is advocated.
• Hempel's Raven Paradox (how observing a non-black non-raven confirms "all ravens are black") is resolved by quantitative Bayesian confirmation, which shows such evidence offers extremely weak confirmation compared to observing a black raven.
• Argument strength should be assessed with a two-dimensional theory: one dimension is the probability of the conclusion given the premises, and the other is the relevance (or confirmation) of the premises to the conclusion.
• The design of experiments, particularly their "confirmational power," plays a crucial role in establishing scientific consensus, with robust experiments overwhelming prior beliefs more easily than those yielding weaker evidence.

Quotes

At 0:03 - "One of the things that I always like to say about science and how it gets done is that science never proves things." - Sean Carroll introducing a key distinction about scientific methodology. At 9:43 - "The frequency theory... Frequencies are just the way we may be know about probabilities, but they're not what the probabilities are." - Branden Fitelson critiquing the frequency theory of probability. At 21:33 - "Hume points out that, well, that argument assumes some kind of principle of regularity of nature... Now if you ask how are you going to justify that premise... it's just a circular argument." - Branden Fitelson explaining Hume's problem of induction. At 36:03 - "'Everyone's entitled to their own priors, no one's entitled to their own likelihoods.'" - Sean Carroll paraphrasing a common Bayesian saying that Branden Fitelson agrees with. At 56:22 - "I'm offering a two-dimensional theory of argument strength. For an argument to be strong, it's got to be probable, sure, yeah, it should be more probable than not given the premise... But also, the evidence should be relevant." - Branden Fitelson proposing his two-dimensional theory of argument strength.

Takeaways

• Embrace that science doesn't "prove" but rather builds confidence through inductive reasoning and probabilistic confirmation, always subject to revision.
• Differentiate clearly between subjective prior beliefs and objective likelihood ratios (Bayes factors), recognizing the latter as the invariant, objective information scientists should report and focus on for evaluating evidence.
• Understand that a robust "logic of confirmation" exists, allowing for the quantitative assessment of how strongly evidence supports a hypothesis, providing a tool to resolve philosophical paradoxes like Hempel's Raven Paradox.
• When evaluating arguments or evidence, consider a two-dimensional approach: both the probability of the conclusion given the premises and the relevance or confirmational power of the evidence itself.
• Recognize that the strength of experimental design, specifically its "confirmational power," significantly influences the role of prior beliefs and the speed of scientific consensus in different fields.