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arxiv: 2607.00034 · v1 · pith:EVWILRD6new · submitted 2026-06-24 · 💻 cs.LO · cs.FL· cs.SY· eess.SY· math.PR

Bayesian updates from coalgebraic determinisation

Manuel Baltieri , Nathaniel Virgo This is my paper

Pith reviewed 2026-07-02 21:18 UTC · model grok-4.3

classification 💻 cs.LO cs.FLcs.SYeess.SYmath.PR
keywords coalgebraic determinisationunifilarisationBayesian filteringMealy machinesmonadsstochastic automatacausal behavioursPOMDPs
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The pith

Unifilarisation of stochastic Mealy machines arises as an instance of coalgebraic determinisation, yielding causal stochastic behaviours via Bayesian updates.

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

The paper shows that unifilarisation, which converts a stochastic Mealy machine into one whose states are priors over the original states and whose transitions perform Bayesian filtering, is a special case of the general coalgebraic determinisation construction. This construction is applied to Mealy machines over monads that carry extra structure generalising the support of a distribution. The resulting semantics assigns to each finite input word a full distribution over output words that respects causality, rather than only a distribution over the current output. A sympathetic reader would care because this supplies a uniform categorical account of the finer observation-conditioning semantics needed by agents in partially observable settings.

Core claim

Unifilarisation arises from the general determinisation procedure applied to Mealy machines over monads equipped with extra structure generalising the notion of the support of a distribution. In this setting the construction produces the Bayesian update maps, and the induced final coalgebra semantics consists of causal stochastic behaviours: families that map input words to distributions on output words compatible with the constraint that outputs cannot depend on future inputs.

What carries the argument

The coalgebraic determinisation construction on structured coalgebras for Mealy machines over monads with generalised support structure, which directly supplies the Bayesian filtering transitions of unifilarisation.

If this is right

  • The Bayesian update maps appear directly as the transitions produced by the general determinisation procedure.
  • The final semantics is strictly finer than the Moore-style semantics whenever intermediate observations must be conditioned upon.
  • Causal stochastic behaviours are obtained uniformly for any monad carrying the required extra structure.
  • The same construction supplies a semantics appropriate for agents that plan by conditioning on entire observation sequences.

Where Pith is reading between the lines

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

  • The unification suggests that other coalgebraic constructions and algorithms could be ported to Bayesian filtering without ad-hoc definitions.
  • The approach could be tested by verifying that common distribution monads satisfy the extra structure and recover the usual filtering equations.
  • Extensions to infinite-state or continuous-state machines would require checking whether the same monad structure still induces valid Bayesian updates.

Load-bearing premise

The monads must carry extra structure generalising the support of a distribution so that determinisation produces the Bayesian update maps.

What would settle it

A concrete monad equipped with the generalised support structure for which the determinisation construction fails to reproduce the known Bayesian filtering transitions on a small stochastic Mealy machine.

read the original abstract

The powerset construction is the classical determinisation procedure for nondeterministic finite automata. In the coalgebraic setting, this construction has been generalised to structured coalgebras, which are coalgebras equipped with extra data. For stochastic Moore machines over the distribution monad, a type of structured coalgebra, the determinisation construction induces a semantics assigning to each finite input word a distribution on the current output. This semantics is appropriate when only the current output matters, but it is too coarse for settings in which intermediate observations must also be taken into account, as is typical for agents solving POMDPs in control theory and reinforcement learning. In these contexts, agents need to condition on all realised observations, not just the final one, so to better plan for the future. This has been addressed from a category theoretic perspective through a procedure called ``unifilarisation'', which (in our context) takes a stochastic Mealy machine and produces a machine whose states are priors over the original state space and whose transitions are given by Bayesian filtering. Here we show that unifilarisation is an instance of coalgebraic determinisation. We work with Mealy machines over monads equipped with extra structure generalising the notion of the support of a distribution. We show that in this setting, unifilarisation arises from the general determinisation procedure. We then compare the resulting final coalgebra semantics with the Moore-style one. Instead of assigning only a distribution on current outputs to each finite input word, it yields causal stochastic behaviours, that is, families mapping input words to distributions on output words compatible with the ``causality'' constraint that outputs cannot depend on future inputs.

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

0 major / 3 minor

Summary. The paper claims that unifilarisation of stochastic Mealy machines arises directly as an instance of the coalgebraic determinisation construction once the underlying monads are equipped with additional structure that generalises the support of a distribution. In this setting the determinisation produces machines whose states are priors over the original state space and whose transitions implement Bayesian filtering; the induced final-coalgebra semantics therefore assigns to each finite input word a distribution over output words that respects the causality constraint (outputs depend only on past and present inputs), in contrast to the coarser Moore-style semantics that records only a distribution on the current output.

Significance. If the identification holds, the result supplies a uniform categorical account of Bayesian updating inside the existing coalgebraic determinisation framework. This unification is potentially useful for transferring results between the two literatures and for giving a coalgebraic semantics to the causal stochastic processes that arise in POMDPs and reinforcement learning. The explicit comparison between the causal and Moore-style final semantics is a concrete contribution that clarifies the difference in granularity.

minor comments (3)
  1. The abstract states that the extra monad structure 'generalises the notion of the support of a distribution,' but the precise axioms imposed on this structure (and the verification that they are satisfied by the distribution monad) should be stated explicitly in the main text before the determinisation construction is applied.
  2. The comparison between the causal semantics and the Moore-style semantics (final paragraph of the abstract) would benefit from a short table or diagram that records, for a small example machine, the two different maps induced by a given input word.
  3. Notation for the monad, the extra structure, and the resulting coalgebra morphisms should be introduced once and used consistently; several distinct symbols appear to be used for the same Bayesian-update map in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report accurately captures the main contribution: showing that unifilarisation arises as an instance of coalgebraic determinisation once monads are equipped with support structure, yielding causal rather than Moore-style semantics.

Circularity Check

0 steps flagged

No significant circularity; direct specialization of general construction

full rationale

The paper equips monads with extra structure that generalizes the support of a distribution, then shows by direct comparison that the existing coalgebraic determinisation procedure applied to Mealy machines over these monads yields exactly the Bayesian filtering maps of unifilarisation. This is an explicit instance relation rather than a redefinition, fit, or self-referential construction. No load-bearing self-citation, uniqueness theorem, or ansatz smuggling is indicated in the provided text; the central claim reduces to verifying that the general procedure reproduces the target maps once the structure is present. The derivation is therefore self-contained against the external coalgebraic framework.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the standard coalgebraic framework for structured coalgebras and on the existence of an additional operation on the monad that generalises support; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Monads carry an extra operation generalising the support of a distribution
    Invoked to make the determinisation construction coincide with Bayesian filtering (abstract).
  • standard math Standard properties of coalgebras and determinisation hold for the structured case
    Background assumption of the coalgebraic setting used throughout.

pith-pipeline@v0.9.1-grok · 5836 in / 1324 out tokens · 25493 ms · 2026-07-02T21:18:48.835167+00:00 · methodology

discussion (0)

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