In Defense of Bayes

Answering the critical rationalists

Ask a certain kind of Bayesian how confident he is that evolution by natural selection is true, and he will give you a number — 0.99999, say — as if he had just read a very reliable weather forecast. Ask the critical rationalists what they make of that number, and they will tell you it is not merely wrong but meaningless. David Deutsch, and Brett Hall following him, mount the most serious standing attack on Bayesianism in circulation: induction is a myth, theories are not the kind of thing that has a probability, and Bayesian epistemology — which appears to restore induction through arithmetic by assigning probabilities to theories — is therefore rotten at the root.

The critics identify a real error but locate it in the wrong place. The number attached to evolution is not meaningless; it is mislabeled. Getting the label right — seeing what the number is about — rescues Bayes from the critique while preserving its valid objections. What Knowledge Is drew the line between explanatory and empirical knowledge and deferred the question of whether explanatory theories can bear credences. Here is the answer.

The Critics’ Case

The attack has two prongs.

First, deductive entailment cannot justify induction by itself. No number of white swans entails all swans are white. Popper, Deutsch, and Hall emphasize that empirical theories remain conjectural and must face criticism, testing, and possible refutation. Bayesian confirmation addresses a different question: how evidence should change comparative confidence given an explicit model. It does not turn finite observations into deductive proof.

Second, theories are not probabilistic. What would it even mean to say general relativity has a 90 percent probability of being true? A coin has a probability of landing heads because there is a chance process behind it — an ensemble of outcomes with weights. There is no chance process behind general relativity. It is not one draw from an urn of possible physics. It either correctly describes the structure of spacetime or it does not, and no experiment anywhere samples from a distribution over its truth. A probability assigned to a theory refers to nothing.

Bayesianism, as the critics read it, commits both sins at once. Its priors over theories are arbitrary numbers with no objective referent, and its updating rule is induction resurrected: every confirming observation nudges the theory’s probability upward, as if justification accumulated by increments. They will grant that scientists reason under uncertainty — their own method demands it — but the reasoning they endorse is abduction, inference to the best explanation: propose conjectures, evaluate them by explanatory power and resistance to refutation, and provisionally adopt the best one, holding it not as probable but as undefeated. Abduction and Bayes look superficially alike — both are ampliative, both weigh rival hypotheses — but abduction assigns no priors, computes no numbers, and claims no incremental justification. On the critics’ map, that is the whole difference between honest conjecture and inductivist bookkeeping.

What They Get Right

There is real ground to give.

Bayes’ theorem contains no complete mechanism for generating hypotheses; it redistributes belief over a model space that must be created, revised, and sometimes expanded. A Bayesian who thinks updating alone explains where quantum mechanics came from has mistaken an accounting rule for a research program. Creative conjecture and criticism are indispensable, while Bayesian comparison can still help assess conjectures once likelihoods and priors have been stated.

Theories also need not carry objective physical probability. In the optional QBU model, Measure is a weight over specified record sectors, not over rival theories. A posterior probability for a theory is therefore an epistemic quantity relative to a model class, prior, and evidence—not a physical field attached to the theory. Deutsch and Hall are right to reject that reification.

But two errors keep the critics from stopping where their argument stops.

The First Error: Theories Are Not Binary

The critique needs theories to be binary — flatly true or flatly false — so that any number strictly between zero and one marks a confusion. They are not binary. Newtonian mechanics is not false, full stop; it is valid to extraordinary accuracy within its domain and fails outside it. It was superseded by relativity and quantum mechanics, yet we still build bridges and fly spacecraft with it, and we are right to. A theory is contextually and approximately true — true given a domain, a scale, a tolerance — which is precisely the conditional structure that all truth turns out to have. “Is Newtonian mechanics true?” is not a well-formed yes-or-no question until the conditions are filled in.

Once theories have scope, accuracy, and limits, there are graded facts about them to be uncertain of. My uncertainty about a theory is rarely uncertainty about a single bit; it is uncertainty about where its domain ends, how good its approximations are, which of its idealizations will crack first. Those are exactly the kinds of questions that admit degrees of belief, because they resolve piecewise against evidence. The binary picture of theories — the picture the critique borrows without argument — repeats the absolutism about truth this volume rejected in its opening chapters.

The Second Error: Probability In, Credence About

The deeper error is a conflation, and undoing it is the hinge of the whole defense.

There are two different things a probability attached to a theory could mean. It could be a probability in the theory’s subject matter — an objective chance, a Measure, a claim that reality itself is running a weighted process over the theory’s truth. That is what the critics attack, and they are right: no such chance exists. Or it could be a Credence about the theory — a quantified description of my epistemic state, my rationally managed uncertainty as to whether, and where, and how well the theory describes reality. That is a different claim about a different subject. The first puts the probability in the world; the second puts it in the believer, where the uncertainty lives.

Deutsch and Hall demolish the first reading and then, without noticing the seam, extend the demolition to the second. But the second does not depend on the first. A Credence about a theory does not attribute partial truth or objective chance to the theory, any more than my uncertainty about the thousandth digit of \(\pi\) attributes fuzziness to arithmetic. The digit is what it is, necessarily; I am the one who doesn’t know it — and my not-knowing is real, graded, and consequential, so it had better be managed coherently. Garrabrant’s Logical Induction makes the point rigorous: coherent credences can and must be assigned even to purely logical statements, where objective probability is not just absent but impossible — a result whose implications for the varieties of credence I develop in the varieties of uncertainty. If credences are legitimate about necessary truths, the absence of a chance process behind general relativity is no obstacle to credences about general relativity.

Quantification is valuable when the problem supports it, but false precision is not a virtue. Agents still choose experiments, designs, and treatments under uncertainty, and explicit weights can expose inconsistencies. Yet choices need not determine a unique precise probability without assumptions about preferences and the betting setup. Qualitative or interval judgments can be more honest when evidence is sparse.

So the useful boundary is this: probabilities posited within a theory concern its subject matter; Credences about a theory describe an agent’s uncertainty. The latter are legitimate and often useful when their conditions are explicit, but they need not always be precise numbers.

The Division of Labor

Once the conflation is undone, the result is not a truce between Bayesians and critical rationalists but a division of labor that each side has been mistaking for a turf war.

Abduction is central to theory creation. Explanatory theories are conjectured rather than mechanically read from data; they are judged by empirical adequacy, explanatory coherence, simplicity, scope, and resistance to criticism. Bayesian methods can guide search and model expansion, but a fixed prior does not exhaust scientific creativity.

Bayes governs a major class of belief-revision problems. Given a stable model space and a specified evidence model, conditionalization says how Credence should change. When evidence is ambiguous, priors are imprecise, or new explanations alter the space itself, further judgment and robustness analysis are required. If QBU is adopted, some empirical uncertainty can be represented as self-location among weighted records; weather, genetic tests, and historical inference can also be analyzed without that ontology. One set of tools helps build and criticize maps; another helps allocate confidence among their consequences.

The two jobs interlock without competing. Abduction proposes; criticism culls; Bayes keeps the books — on your branch-location within the surviving theories’ structure, and on the theories themselves, whose scope and accuracy are things you can be rationally more or less sure of. Even the drama of theory choice, which both camps claim, splits cleanly along the seam: the search for a better explanation is abductive, while your confidence that the current best explanation will survive its next test is a credence, revisable in the ordinary Bayesian way. This seam has a name, which I draw all the way across at the end of the volume: Bayesian about updating, pluralist about probability, critical-rationalist about knowledge — the stance I call fallibilist Bayesianism.

So the answer deferred from the knowledge chapter is conditional: theories can bear Credence without carrying an objective physical probability. Science is not a scoreboard of automatically rising confirmation, but action still requires comparative uncertainty. Bayes is not a complete theory of where knowledge comes from. It is one disciplined way to revise graded uncertainty, best used alongside model criticism, sensitivity analysis, and willingness to enlarge the space of possibilities.