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arxiv: 1907.09472 · v1 · pith:75A7JRZFnew · submitted 2019-07-22 · 💻 cs.AI · cs.LO

Learning Probabilities: Towards a Logic of Statistical Learning

Alexandru Baltag , Soroush Rafiee Rad , Sonja Smets This is my paper

Pith reviewed 2026-05-24 18:30 UTC · model grok-4.3

classification 💻 cs.AI cs.LO
keywords imprecise probabilitiesplausibility mapsstatistical learningbelief revisionalmost sure convergencedoxastic logicsamplingBayes rule
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The pith

Agents learn unknown probabilities by updating plausibility rankings on a fixed set of measures, with beliefs converging almost surely to the true value.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models situations of radical uncertainty about probabilities by combining a set of possible probability measures with a plausibility map that ranks them. Sampling updates only the plausibility map through a plausibilistic version of Bayes' rule, without shrinking or expanding the set of measures. Higher-order information instead reduces the set while leaving plausibilities fixed. Beliefs are identified with what holds in the most plausible measures, and this setup produces non-AGM belief change. The key result is that repeated sampling makes the agent's beliefs converge almost surely to the actual probability.

Core claim

The central claim is that the beliefs obtained by repeated sampling converge almost surely to the correct belief in the true probability, because the sampling update revises only the plausibility map while leaving the given set of measures unchanged.

What carries the argument

Plausibility map over a set of probability measures, updated by a plausibilistic Bayes rule that re-ranks measures without altering the set.

If this is right

  • Belief change from sampling violates the standard AGM axioms.
  • Higher-order linear inequalities shrink the set of measures without affecting their plausibility ordering.
  • Beliefs are defined as propositions true in all most-plausible measures.
  • The model supports a dynamic doxastic logic that combines sampling and higher-order updates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same separation of set contraction and plausibility re-ranking could be tested in sequential decision tasks where agents must act before full convergence occurs.
  • If plausibility is defined via entropy or center-of-mass, the convergence speed may vary with the choice of ranking function, providing a testable prediction for simulations.
  • The framework suggests a way to reconcile imprecise probabilities with pointwise learning without forcing the agent to adopt a single prior at the outset.

Load-bearing premise

The update induced by sampling is a plausibilistic version of Bayes' Rule that changes only the plausibility map while leaving the given set of measures unchanged.

What would settle it

A sequence of independent draws from a fixed distribution in which the most plausible measure after many samples fails to approach the empirical frequencies obtained from those draws.

read the original abstract

We propose a new model for forming beliefs and learning about unknown probabilities (such as the probability of picking a red marble from a bag with an unknown distribution of coloured marbles). The most widespread model for such situations of 'radical uncertainty' is in terms of imprecise probabilities, i.e. representing the agent's knowledge as a set of probability measures. We add to this model a plausibility map, associating to each measure a plausibility number, as a way to go beyond what is known with certainty and represent the agent's beliefs about probability. There are a number of standard examples: Shannon Entropy, Centre of Mass etc. We then consider learning of two types of information: (1) learning by repeated sampling from the unknown distribution (e.g. picking marbles from the bag); and (2) learning higher-order information about the distribution (in the shape of linear inequalities, e.g. we are told there are more red marbles than green marbles). The first changes only the plausibility map (via a 'plausibilistic' version of Bayes' Rule), but leaves the given set of measures unchanged; the second shrinks the set of measures, without changing their plausibility. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds. But our belief change does not comply with standard AGM axioms, since the revision induced by (1) is of a non-AGM type. This is essential, as it allows our agents to learn the true probability: we prove that the beliefs obtained by repeated sampling converge almost surely to the correct belief (in the true probability). We end by sketching the contours of a dynamic doxastic logic for statistical learning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a model for beliefs about unknown probabilities under radical uncertainty by equipping a set of probability measures with a plausibility map. It distinguishes two learning operations: type (1) sampling updates only the plausibility map via a plausibilistic Bayes rule while leaving the set of measures fixed; type (2) higher-order linear inequalities shrink the set without altering plausibilities. Beliefs are defined as properties true in the most plausible measures (in the style of belief revision). The central claim is a proof that repeated type-(1) updates yield beliefs that converge almost surely to the true probability; the paper notes that the induced revision violates standard AGM axioms, which is presented as essential for the convergence result, and sketches a dynamic doxastic logic for statistical learning.

Significance. If the convergence result holds, the framework provides a technically novel bridge between imprecise probabilities and non-AGM belief revision that permits agents to learn the true measure from sampling. The explicit use of a fixed set plus plausibility map, together with the non-AGM revision operator, is a substantive contribution to the logic of statistical learning and could inform work in AI epistemology and dynamic epistemic logic.

major comments (2)
  1. [Abstract / learning type (1) paragraph] Abstract and the paragraph describing learning type (1): the convergence claim ('beliefs obtained by repeated sampling converge almost surely to the correct belief (in the true probability)') is stated without any mention that the true measure must belong to the initial fixed set of measures. Because type-(1) learning is defined to leave the set unchanged, convergence to the true measure is impossible unless that measure is already present; this assumption is load-bearing for the central theorem yet is neither stated nor justified.
  2. [Model definition and learning type (1)] The definition of beliefs (most plausible measures within the fixed set) together with the invariance of the set under sampling implies that the model can only ever represent beliefs inside the initial set. The manuscript should therefore either add the explicit hypothesis that the true measure lies in the initial set or qualify the convergence statement to 'convergence to the most plausible measure within the initial set that is closest to the true one.'
minor comments (1)
  1. [Introduction / model section] The examples of plausibility maps (Shannon Entropy, Centre of Mass, etc.) are listed without formal definitions or citations; adding precise mathematical statements would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify an implicit modeling assumption that was not made explicit in the abstract or model description. We will revise the manuscript to address both points.

read point-by-point responses

  1. Referee: [Abstract / learning type (1) paragraph] Abstract and the paragraph describing learning type (1): the convergence claim ('beliefs obtained by repeated sampling converge almost surely to the correct belief (in the true probability)') is stated without any mention that the true measure must belong to the initial fixed set of measures. Because type-(1) learning is defined to leave the set unchanged, convergence to the true measure is impossible unless that measure is already present; this assumption is load-bearing for the central theorem yet is neither stated nor justified.

    Authors: We agree that the assumption is load-bearing and was not stated. The framework is designed for agents whose initial set of measures includes the true probability (as is standard when modeling radical uncertainty over an unknown distribution). We will revise the abstract and the type-(1) description to state this hypothesis explicitly and note that it is required for the almost-sure convergence result. revision: yes

  2. Referee: [Model definition and learning type (1)] The definition of beliefs (most plausible measures within the fixed set) together with the invariance of the set under sampling implies that the model can only ever represent beliefs inside the initial set. The manuscript should therefore either add the explicit hypothesis that the true measure lies in the initial set or qualify the convergence statement to 'convergence to the most plausible measure within the initial set that is closest to the true one.'

    Authors: We agree that beliefs remain inside the initial set. We will add the explicit hypothesis that the true measure belongs to the initial set (rather than qualifying the convergence claim), as this matches the intended modeling choice: the agent considers a set of possible measures that includes the unknown true probability. The revision will appear in the model-definition section and will be cross-referenced in the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity; convergence is a stated theorem under standard model assumptions

full rationale

The paper defines a fixed set of probability measures that sampling does not alter, updating only a plausibility ordering via a non-AGM rule. The central result is an almost-sure convergence theorem for beliefs (defined via most-plausible measures) to the true probability. This is presented as a proof relying on external measure-theoretic probability, not as a re-derivation or fit of the input data. No equation reduces the claimed limit to a parameter fitted from the same data, no self-citation supplies a uniqueness theorem that forces the result, and the fixed-set modeling choice is an explicit modeling decision rather than a definitional loop. The derivation therefore remains independent of its target conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the model rests on the existence of a plausibility map and a specific non-standard update rule whose justification is not supplied.

axioms (1)
  • domain assumption Plausibilistic version of Bayes' Rule updates only the plausibility map while the set of measures stays fixed
    Stated in the abstract as the mechanism for type-(1) learning.

pith-pipeline@v0.9.0 · 5838 in / 1179 out tokens · 23930 ms · 2026-05-24T18:30:11.533435+00:00 · methodology

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