Did Karl Popper argue against Bayesian inference?

Question

I am somewhat familiar with the works of Karl Popper and his opposition to using past data to induce prospects of future events, however disclaimed as uncertain, AKA historicism. He contributed to formulating the problem of induction, which IMHO conflates the certainty of an outcome with a rational probability distribution of possible outcomes based on slow changing detectable patterns, invalidating the latter for the absence of the former, a form of strawman fallacy.

I'm not sure if it's been formally recognized as such but Baysian inference can be thought of as realistically, however partially, historicist and inductive. Does Popper oppose the general validity of Baysian thought on the grounds of being inductive and historicist, perhaps not exact enough for his standards, although it never lays such claims as it clearly deals in probabilistic (approximate) and not exact terms?

Answer 1

Bayesian inference seeks to believe in that which has a high conditional probability as computed with Bayes's theorem. The problem with using $P\left( A|B\right)=P\left( B|A\right)\frac{P\left( A\right)}{P\left( B\right)}$ to compute the LHS is that, although the first factor on the RHS is often known, at least one of the RHS's other probabilities usually won't be known. (If $A$ denotes a theory and $B$ data, $P\left( B\right)$ requires another theory to compute (a null hypothesis), and $P\left( A\right)$ is even harder.) If you want to see Popper's arguments that (a) the probability of a theory given data will, counter-intuitively, be $0$ and (b) probability is the wrong way to think about the case for or against a theory, but (c) a non-zero measure expressible in terms of conditional probability called verisimilitude is more relevant, see Objective Knowledge: An Evolutionary Approach.

Historicism doesn't mean “induction works”. As you can see in the Wikipedia article you linked to, Popper took the term to mean political theories that imply human society will inevitably turn out a certain way. In The Poverty of Historicism, Popper critiqued such theories (e.g. Marxism claiming capitalism will be overthrown) on the grounds that the destiny of society depends on its future discoveries, which by definition cannot be foreseen.

Popper's main contribution regarding induction was not to critique it (though he did explain how the existing criticism had been misunderstood), but to argue that science didn't need it so was more justifiable than anti-inductivists had thought. See The Logic of Scientific Discovery. Hume formulated the first problem of induction, though Goodman and Popper each cited additional reasons it's a problematic way to reason.

You can claim the problem “conflates the certainty of an outcome with a rational probability distribution of possible outcomes”, but it's important to understand that, while inductivists of the time such as Carnap and Russell felt induction and the problem thereof were that straightforward, Popper argued they'd misunderstood the problem. I'll summarize it in the hope its beyond-mere-uncertainty implications will become clear. Hume argued that, because a principle of uniformity of nature is (a) required to find induction convincing and (b) unknowable a priori because it's not a tautology, the only way it could be justified is empirically. But evidence that nature fits such a principle would by definition be presented in an inductive argument! As Popper noted in (if memory serves) Conjectures and Refutations, this forces us to either (1) assume such a principle in the argument that justifies it (circularity) or (2) use a different principle that only pushes the problem back (infinite regress).

Answer 2

The answer is not so straightforward. Of course, the thrust of Popper's position was against probabilistic induction in general, and Bayesianism is often put forth as the leading alternative to falsificationism, see Hypothesis testing: Fisher vs. Popper vs. Bayes. In 1983 Popper (with Miller) even offered a technical argument published as a note A proof of the Impossibility of Inductive Probability in Nature, which concludes with:

"This result is completely devastating to the inductive interpretation of the calculus of probability. All probabilistic support is purely deductive: that part of a hypothesis that is not deductively entailed by the evidence is always strongly countersupported by the evidence-the more strongly the more the evidence asserts. This is completely general; it holds for every hypothesis h ; and it holds for every evidence e, whether it supports h, is independent of h, or countersupports h. There is such a thing as probabilistic support; there might even be such a thing as inductive support (though we hardly think so). But the calculus of probability reveals that probabilistic support cannot be inductive support."

The Popper-Miller argument is controversial, and many consider it invalid, see commentary at Intellectual Mathematics, but the sentiment seems clear. However, in Induction versus Popper: substance versus semantics Greenland argues that "some of the controversy is semantic, and hence Popperian criticisms of induction must be translated carefully into ordinary language to be appreciated by inductively oriented epidemiologists." In other words, Popper's sense of "induction" is not the same as colloquially used, or as understood in Bayesianism and other formalized approaches:

"Nearly every type of statistical approach has at times been called ‘inductive’. R. A. Fisher referred to his significance-testing procedures as methods for ‘inductive inference’; Neyman referred to his hypothesis-testing procedures as methods for ‘inductive behavior’; R. von Mises referred to his objective Bayesian approach as an ‘inductive science’; and DeFinetti referred to his subjective Bayesian approach as ‘inductive reasoning’. Despite the diversity of approaches represented by these authors, their usage is defensible."

This leaves room for some conciliation. The Fisher-Neyman significance testing is usually endorsed by falsificationists, and in his own quantitative accounts of corroboration Popper used Bayesian looking measures. Indeed, the controversy seems to be chiefly over interpretation of formalisms, including the Bayesian one, rather than their potential usefulness in specific contexts.