Conditionalization and Reflection

Probabilism constrains credence at a time. It says nothing about how credence should change, and a rational agent must learn. The standard dynamic norm is conditionalization: on acquiring evidence $E$ and nothing more, replace the old credence function with the old function conditioned on $E$.

$$C{r}_{\text{new}}(A)=C{r}_{\text{old}}(A\mid E)=\frac{C{r}_{\text{old}}(A\wedge E)}{C{r}_{\text{old}}(E)}$$

This is the engine of Bayesian epistemology and the reason the framework is called Bayesian at all: Bayes' theorem is a trivial consequence of the axioms, but the rule that belief should update by conditioning on evidence is a substantive normative claim requiring its own defence.

This page sets out the rule, the generalisation needed when evidence is uncertain, the arguments for it, the reflection principles that govern an agent's attitude to their own future credences, and the two places where the framework strains: zero priors and new hypotheses.

What the rule requires

Three features of conditionalization deserve to be made explicit, because each is a substantive commitment rather than a formality.

It is defined only when $C{r}_{\text{old}}(E)>0$. An agent who assigned zero to $E$ cannot conditionalize on it — the ratio is undefined — and so cannot learn it however overwhelming the evidence. This is the motivation for regularity and Cromwell's rule, discussed under probabilism and again below.

It requires learning $E$ with certainty. After the update, $C{r}_{\text{new}}(E)=1$, and since conditionalization can never lower a credence of $1$, the agent is permanently committed. Real observation rarely warrants this. Glimpsing a bird in poor light does not make "that was a finch" certain, and treating it as certain is a modelling error with permanent consequences.

It requires that $E$ be the agent's total evidence. Conditioning on part of what one learns can move credence in the wrong direction, so the rule presupposes that the evidence can be captured by a single proposition — which is exactly what the next section relaxes.

Jeffrey conditionalization

Richard Jeffrey's generalisation handles uncertain evidence. Instead of learning a proposition with certainty, experience redistributes credence across a partition $\{E_1,\dots ,E_n\}$, moving the credences to new values $q_i$. The new credence in any $A$ is then the weighted average of the conditional credences:

$$C{r}_{\text{new}}(A)=\sum _i{q}_i\cdot C{r}_{\text{old}}(A\mid E_i)$$

Strict conditionalization is the special case where some $q_i=1$. The rule preserves the conditional probabilities $Cr(A\mid E_i)$ — this is its defining feature, called rigidity — while allowing the partition's probabilities to shift.

Jeffrey's motivation was the observation that experience does not hand us propositions. Looking at a cloth by candlelight might shift credences over $\{$green, blue, violet$\}$ to $0.7,0.25,0.05$ without making any of them certain, and there may be no proposition one learns with certainty that captures what happened.

Two costs come with the generality. Jeffrey updating is non-commutative: updating on one partition and then another generally gives a different result from the reverse order, so the agent's final state depends on the sequence of experiences and not only on their total content. And the rule requires the new values $q_i$ as inputs — it says how to propagate them, not where they come from — so it describes the structure of uncertain learning without explaining it.

Arguments for the rule

Diachronic Dutch book. An agent who announces a policy of updating other than by conditionalization can be offered a timed sequence of bets guaranteeing loss. The weaknesses of this argument are set out under Dutch books: it requires the policy to be known in advance, and it cannot distinguish irrational drift from rational reconsideration.

Expected accuracy. Greaves and Wallace show that, evaluated from the agent's current perspective, the update rule maximising expected accuracy is conditionalization. This is the stronger argument, and it parallels the accuracy-based defence of probabilism, but note that it evaluates rules by the agent's present lights — an agent who suspects their current credences are badly wrong is not obviously bound by it.

Conservatism. Conditionalization is the update that changes the agent's opinions as little as possible while accommodating the evidence: it minimises relative entropy from the old function subject to the constraint $Cr(E)=1$. This gives a satisfying characterisation, though "change as little as possible" is a methodological preference standing in need of its own justification.

Reflection

If conditionalization governs how credences evolve, what attitude should an agent take toward their own future credences? Van Fraassen's Reflection Principle answers:

$$C{r}_{\text{now}}(A\mid C{r}_{\text{later}}(A)=x)=x$$

An agent who expects to have credence $x$ in $A$ tomorrow should have credence $x$ in $A$ today, conditional on that expectation. The rationale is that one should treat one's future self as an expert, since they will have strictly more evidence.

There is an elegant formal connection here. An agent who conditionalizes has credences that form a martingale: the expectation of tomorrow's credence, conditional on today's information, equals today's credence. This is exactly the martingale property, and Lévy's upward convergence theorem then guarantees that credences converge as evidence accumulates. Reflection is thus not an ad hoc addition but a consequence of conditionalization for an agent certain they will conditionalize.

That proviso is where the objections enter, and they are serious:

• Anticipated irrationality. Someone who knows they will be drunk, drugged, or brainwashed tonight should not defer to their future credences. Reflection as stated makes such deference mandatory.
• Memory loss. An agent expecting to forget evidence should not defer to a future self who knows less. This is the mechanism behind the Sleeping Beauty problem and the self-locating puzzles generally.
• Self-fulfilling and self-referential cases generate paradoxes familiar from other principles that require deference to an expert who is oneself.

The natural repair restricts Reflection to agents who will remain rational and lose no evidence. That is defensible but reveals the principle's real content: it is a consequence of conditionalization under ideal conditions rather than an independent constraint, and it inherits every idealisation of the underlying framework.

Where the framework strains

Zero priors

Conditionalization cannot resurrect a hypothesis assigned credence zero. If a scientist's prior over theories excludes the true one, no evidence can help, and the framework offers no remedy. Regularity — assign zero only to logical falsehoods — is the obvious fix and is unavailable in continuous spaces where individual outcomes must receive zero. The infinitesimal repair using non-standard probability is technically available and philosophically contested.

The practical version of this is more troubling than the formal one. Credences that are not zero but merely very small behave almost as badly: with a prior of $10^{-30}$, a hypothesis needs overwhelming evidence to become live, and an agent can be effectively closed-minded while remaining formally regular.

New hypotheses

The deeper problem. Conditionalization operates on a fixed algebra of propositions. It says what to do when the evidence changes and nothing at all about what to do when the hypothesis space changes — when a theory nobody had conceived is formulated for the first time.

This is not a marginal case; it is the central event in scientific revolutions. Before 1915 no one had a credence in general relativity, because the theory did not exist to be believed. The Bayesian framework has no rule for introducing it, and doing so requires redistributing credence over an expanded algebra in a way the axioms do not license.

Proposed treatments — reserving credence for a catch-all "some theory not yet considered", or modelling the agent as logically non-omniscient with an implicit algebra made explicit over time — are active research rather than settled doctrine. This is the same idealisation that generates the problem of old evidence, and both are consequences of modelling agents as logically omniscient over a fixed space of propositions, an idealisation whose costs models and applications flagged in general terms.

Where this sits

Conditionalization is the second of the two Bayesian norms, and the less secure. Probabilism has two independent arguments; conditionalization has a weak diachronic Dutch book, an accuracy argument that is perspective-relative, and a conservatism rationale that is more a description than a justification.

Together the two norms constrain the structure and the dynamics of credence, and leave the starting point entirely open. Every substantive disagreement in Bayesian epistemology — between subjectivists and objectivists, over the confirmation of theories, over anthropic reasoning — is at bottom a disagreement about priors, which is the next page.