Partial Markov Categories

• Partial Markov categories are copy–discard categories with conditionals that allow non-total morphisms, modeling probabilistic inference with constraints and partiality.
• They support a unified, string-diagrammatic calculus for conditionals, Bayesian inversion, and normalization, ensuring unique almost-sure updates.
• They bridge subprobability models and cartesian restriction methods, enabling synthetic proofs of Bayes’ theorem and abstract probabilistic inequalities.

Partial Markov categories are copy–discard categories with conditionals in which morphisms need not be total, and they were introduced as a synthetic framework for probabilistic inference that combines the synthetic probability of Markov categories with the partial-map perspective of cartesian restriction categories. In this setting, stochastic processes, constraints, observations, updates, and normalization are handled within a single string-diagrammatic formalism; the motivating slogan is that, just as Markov categories encode stochastic processes, partial Markov categories encode stochastic processes with constraints, observations, and updates (Lavore et al., 2023, Lavore et al., 24 Jan 2025).

1. Copy–discard structure and the passage from total to partial

A partial Markov category is built on a copy–discard category $(\mathcal C,\otimes,I)$: every object $X$ carries a commutative comonoid structure

$$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$

compatible with the monoidal product. Unlike in an ordinary Markov category, copy and discard are not required to be natural for all morphisms. This failure of universal naturality is precisely where partiality enters (Lavore et al., 2023).

Within this ambient structure, a morphism $f:X\to Y$ is deterministic when it preserves copying,

$\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$

and total when it preserves discard,

$\varepsilon_Y\circ f=\varepsilon_X.$

A Markov category is then a copy–discard category with conditionals in which all morphisms are total, whereas a partial Markov category drops that totality requirement while retaining conditionals (Lavore et al., 24 Jan 2025).

This shift from totality to partiality has a direct probabilistic interpretation. For a general morphism $f:X\to Y$, the composite $f\circ \varepsilon_Y:X\to I$ is its probability of failure, or, in deterministic language, its domain of definition. The framework isolates a useful intermediate notion, quasi-totality: in a partial Markov category, $f$ is quasi-total iff its probability of failure is deterministic (Lavore et al., 2023). This imports into categorical probability the same kind of domain bookkeeping that restriction categories use for partial deterministic maps.

2. Conditionals, Bayesian inversion, and normalization

The defining analytic strength of partial Markov categories is that they still support conditionals. Given $f:X\to A$ and $X$0, one forms a conditional composition $X$1 by copying the input, feeding one branch through $X$2, and then supplying both the original input and the $X$3-output to $X$4. A conditional of a morphism $X$5 is a quasi-total map $X$6 such that

$X$7

A partial Markov category is exactly a copy–discard category in which such conditionals exist for all $X$8 (Lavore et al., 24 Jan 2025).

Almost-sure equality is internalized synthetically. For $X$9 and $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$0, one declares $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$1 and $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$2 to be $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$3-almost surely equal when

$$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$4

Conditionals, Bayesian inverses, and normalizations are unique only up to this internal almost-sure equality, mirroring measure-theoretic practice (Lavore et al., 24 Jan 2025).

Bayesian inversion is obtained as a special case of conditionalization. For a morphism $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$5 and a prior $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$6, a Bayesian inversion $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$7 exists in any partial Markov category and is unique up to $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$8-almost-sure equality. The same formalism yields a compositional Bayes law for composites: the Bayesian inversion of $$\Delta_X:X\to X\otimes X,\qquad \varepsilon_X:X\to I,$$9 with respect to $f:X\to Y$0 factors through the inversions of $f:X\to Y$1 and $f:X\to Y$2 (Lavore et al., 24 Jan 2025).

Normalization is the other fundamental operation. For $f:X\to Y$3, a normalization $f:X\to Y$4 is characterized by

$f:X\to Y$5

In subprobability semantics, this is exactly the passage from a subdistribution to the corresponding normalized distribution on the support where the total mass is nonzero. Normalizations exist in every partial Markov category and are unique up to almost-sure equality relative to the domain map $f:X\to Y$6 (Lavore et al., 24 Jan 2025). This makes partial Markov categories an intrinsic setting for the common inference pattern “constrain, then renormalize.”

3. Discrete structure, observations, and categorical updating

A further refinement is obtained by adding comparators. A discrete partial Markov category is a partial Markov category equipped with comparator maps

$f:X\to Y$7

such that $f:X\to Y$8 is a partial Frobenius monoid. These comparators act as equality constraints and provide the categorical mechanism for exact observations (Lavore et al., 2023).

This extra structure supports a synthetic theory of observations. In a discrete partial Markov category, a deterministic state $f:X\to Y$9 can be turned into a constraint “observe $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$0” by composing copy, the state $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$1, and the comparator $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$2. More generally, for any Markov category $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$3, the construction $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$4 freely adds generators $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$5 for deterministic observations and yields a partial Markov category in which normalizations and conditionals can still be computed in the original base category (Lavore et al., 2023).

The resulting calculus recovers Bayes’ theorem in a purely structural form. In any discrete partial Markov category, observing a deterministic $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$6 through a channel $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$7 from a prior $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$8 yields, after normalization, the same posterior as evaluating the Bayesian inversion of $\Delta_Y\circ f=(f\otimes f)\circ \Delta_X,$9 with respect to $\varepsilon_Y\circ f=\varepsilon_X.$0 at $\varepsilon_Y\circ f=\varepsilon_X.$1. The framework thereby identifies exact observation and Bayes inversion as two presentations of the same update (Lavore et al., 2023).

The 2025 development extends this inference-oriented perspective: it describes observations, Bayes’ theorem, normalisation, and both Pearl’s and Jeffrey’s updates in purely categorical terms (Lavore et al., 24 Jan 2025). A plausible implication is that partial Markov categories furnish a unified calculus for both exact conditioning and softer evidence update rules, without leaving the ambient categorical language.

4. Examples and constructive routes

Two families of examples dominate the theory. The first comes from Kleisli categories of subprobability-type monads; the second comes from span-style partialization of ordinary Markov categories.

Construction	Morphisms	Notable property
SubStoch	subdistributions on sets	partial Markov category with conditionals (Lavore et al., 2023)
SubBorelStoch	subprobability kernels on standard Borel spaces	partial Markov category via Giry+Maybe (Lavore et al., 2023)
$\varepsilon_Y\circ f=\varepsilon_X.$2 for partializable $\varepsilon_Y\circ f=\varepsilon_X.$3	spans $\varepsilon_Y\circ f=\varepsilon_X.$4 with $\varepsilon_Y\circ f=\varepsilon_X.$5 a deterministic mono	preserves positivity, representability, conditionals, Kolmogorov products, and splittings of idempotents (Mohammed, 5 Sep 2025)

For subdistribution models, the core examples are the Kleisli categories of the finitary subdistribution monad and of the subprobability Giry monad. These directly model partial stochastic kernels: morphisms are allowed to return mass $\varepsilon_Y\circ f=\varepsilon_X.$6, and discard records the missing mass as failure probability (Lavore et al., 2023).

The partialization construction is more structural. A partializable Markov category is a positive Markov category in which pullbacks of deterministic monomorphisms exist and remain deterministic, and tensoring with an arbitrary object preserves monomorphisms of this form. From such a category $\varepsilon_Y\circ f=\varepsilon_X.$7, one forms $\varepsilon_Y\circ f=\varepsilon_X.$8 whose morphisms are spans

$\varepsilon_Y\circ f=\varepsilon_X.$9

with deterministic monic left leg. Composition is by pullback, tensor is computed componentwise on the apices, and the inherited copy–discard maps turn $f:X\to Y$0 into a CD category of partial kernels (Mohammed, 5 Sep 2025).

This span construction recovers familiar partial-kernel categories. For $f:X\to Y$1, a morphism in $f:X\to Y$2 can be identified with a measurable subset $f:X\to Y$3 together with a Markov kernel $f:X\to Y$4; for discrete and semiring-weighted settings one similarly obtains categories of partial stochastic kernels or partial multivalued maps (Mohammed, 5 Sep 2025).

The preservation results are substantial. In $f:X\to Y$5, all morphisms are quasi-total; total morphisms are exactly those in the image of the original $f:X\to Y$6; copyable morphisms are exactly the spans whose right leg is deterministic in $f:X\to Y$7; and the construction preserves positivity, representability, conditionals, Kolmogorov products, and splittings of idempotents (Mohammed, 5 Sep 2025). This makes partialization a systematic passage from ordinary categorical probability to partial categorical probability.

5. Order, comparators, and inequational reasoning

Recent work adds an order-theoretic layer to partial Markov categories. Besides the restriction or normalization order

$f:X\to Y$8

there is a conditional preorder defined by

$f:X\to Y$9

when there exists a witness $f\circ \varepsilon_Y:X\to I$0 such that

$f\circ \varepsilon_Y:X\to I$1

Intuitively, $f\circ \varepsilon_Y:X\to I$2 means that $f\circ \varepsilon_Y:X\to I$3 is obtained from $f\circ \varepsilon_Y:X\to I$4 by conditioning or reweighting with an evidence effect (Lavore et al., 25 Jul 2025).

Every partial Markov category is canonically enriched over preordered sets via this conditional preorder, and the tensor product is monotone with respect to it. Moreover, the enrichment is subunital—every effect $f\circ \varepsilon_Y:X\to I$5 satisfies $f\circ \varepsilon_Y:X\to I$6—and it is minimal among subunital preorder enrichments. In this precise sense, $f\circ \varepsilon_Y:X\to I$7 is the canonical order carried by a partial Markov category (Lavore et al., 25 Jul 2025).

In the main examples, the abstract order collapses to familiar concrete ones. In $f\circ \varepsilon_Y:X\to I$8 and $f\circ \varepsilon_Y:X\to I$9, $f$0 is exactly the usual pointwise order on subprobability kernels; in cartesian restriction categories it coincides with the restriction order; and in cartesian bicategories of relations it recovers relational inclusion (Lavore et al., 25 Jul 2025).

Comparators are tightly controlled by this order. In a copy–discard–compare category, the multiplication $f$1 is the $f$2-least conditional of the copy $f$3, while the corresponding cap $f$4 is the least conditional of the identity. Conversely, under mild order-theoretic hypotheses, least conditionals of copy and identity force the existence of comparators. A CD-category can therefore admit at most one compare structure (Lavore et al., 25 Jul 2025).

The order also supports genuine inequational probability theory. The paper proposes a synthetic Cauchy–Schwarz inequality for discrete partial Markov categories, derives a means inequality from it, and uses these to prove a categorical validity-increase theorem: updating a prior distribution with an evidence predicate increases the likelihood of the evidence in the posterior (Lavore et al., 25 Jul 2025). This turns partial Markov categories into a setting not only for equations of kernels but also for abstract probabilistic inequalities.

6. Position within categorical probability and current directions

Partial Markov categories sit at the intersection of several neighboring theories. One recent taxonomy places mass categories and domain categories between general gs-monoidal categories, Markov categories, and restriction categories, isolating abstract notions of total mass and domain of definition. This provides a broader axiomatic backdrop for partial probabilistic structure, in which partial Markov categories can be read as combining domain-sensitive behavior with probabilistic mass (Cioffo et al., 27 Aug 2025).

The framework also interacts with more specialized constructions. The paper on Gauss constructions and Moore–Penrose inverses does not define partial Markov categories, but it explicitly observes that without Moore–Penrose inverses the Gauss construction would be a natural example in which conditionals exist only when certain linear equations have solutions. This suggests a source of partiality governed by solvability of $f$5 rather than by explicit domain restriction (Comfort et al., 20 Dec 2025).

Other neighboring developments are more syntactic. The free-Markov-category treatment of causal theories and do-calculus is described as highly adaptable to partiality, because its central results are expressed at the level of syntax, copy–discard structure, and conditionals as properties rather than universal assumptions (Yin et al., 2022). Likewise, latent fibrations, though formulated for restriction categories rather than for Markov categories, provide restriction-enriched fibrational machinery for organizing partial maps and reverse constructions. This suggests a possible fibrational semantics for partial probabilistic inference and partial Bayesian inversion (Cockett et al., 2020).

Taken together, these developments indicate that partial Markov categories have become a central framework for categorical probability in the presence of failure, subnormalization, observation, and update. Their current form already supports Bayes’ theorem, normalization, exact observations, Pearl and Jeffrey updates, canonical order-enrichment, and span-based constructions of partial kernels (Lavore et al., 24 Jan 2025, Mohammed, 5 Sep 2025). A plausible implication is that they will continue to serve as the main meeting point between synthetic probability, restriction-theoretic partiality, and compositional inference.