Bayesian Filters via the Stochastic Category

Introduction

Stochastic morphisms, also known as Markov kernels, between measurable spaces were introduced by Lawvere [19] to express stochastic phenomena such as noise and uncertainty in categorical terms, and independently by Čencov [3, 4] from the viewpoint of statistical decision theory. The resulting category, which we call the stochastic category, was interpreted by Giry [14] as the Kleisli category of the probability monad, and its structure was axiomatized by Fritz [10] as Markov categories (see also [12]), in which marginalization, conditioning and independence are formulated through compositions and monoidal products of morphisms. In this framework, arguments are carried out with string diagrams, and classical algorithms such as the Bayesian filter [11, 25] and the smoother [11] can be derived uniformly.

In classical probability theory, Bayesian filters and their continuous time analogues have been studied extensively [1, 7, 16, 18, 23], and filters are naturally used with feedback, i.e. the transitions and observations of the model may depend on the observations made so far, for instance through a control input or through coefficients chosen based on the data [7 §4.2]. Such models are not hidden Markov models in the sense of [11], since the future is not conditionally independent of

Date: November 1, 2026.

2020 Mathematics Subject Classification. Primary: 60G35, Secondary: 60A05, 62F15, 18M05, 93E11, 28A50.

This work is supported in part by RIKEN AIP, as well as JST CREST Grant Number JPMJCR2112 (The Bayes-Duality Project FY2021–2026, led by Emtiyaz Khan). The second author is supported by the RIKEN Special Postdoctoral Researcher Program, FY2025–2027. the past given the current hidden state. They are instead characterized by their conditional laws given the whole past (Proposition 4.2). The purpose of this article is to extend the categorical interpretation of Bayesian filters to this setting, by systematically parameterizing stochastic morphisms, using notation intuitively compatible with classical probability. In categorical terms, fixed parameters are treated in [11] through the parametric Markov categories of [12 §2.2], and inputs appear in [25 Definition 3.1]. Parameterized stochastic morphisms also appear in [13] as transitional random variables. In this article, the parameter space grows with the observations. For simplicity, we work in the stochastic category, the motivating example of a Markov category.

We take the joint law as the primary object. A joint law is a stochastic morphism $\pi: S \to X \times Y$, that is, a family of probability measures $\pi(s)$ on $X \times Y$ parameterized by a measurable space $S$. We write

for the conditional law of $Y$ given $X$ for $\pi$ (see Definition 2.1), a family of regular conditional probabilities for the measures $\pi(s)$, characterized by

The notation makes explicit the joint law and the parameter space, and the factors $X$ and $Y$ play the role traditionally played by random variables, conditioning on $X$ being conditioning on the projection $\operatorname{pr}_{X}$ (see Remark 2.3). The essential point is that $\pi(Y \mid X)$ is again a stochastic morphism, with parameter space $S \times X$, so that further conditional laws can be taken and the rules of conditioning become equalities of stochastic morphisms. For example, the tower property (Theorem 2.7) for $\pi: S \to X \times Y \times Z$ reads

that is, conditioning on $X$ and then on $Y$ is the same as conditioning on $X \times Y$ at once, corresponding to the identity $p(z \mid x,y) = p(y,z \mid x)/p(y \mid x)$ for densities. No reference measure or density is required, and measure theory is confined to the existence and uniqueness of the conditional law.

To see the calculus at work, consider the simplified case of a hidden Markov model without feedback when $S$ is a single point, with initial law $\mu$ on $X_{1}$, transitions $a_{k} : X_{k} \to X_{k+1}$ and observations $h_{k} : X_{k} \to Y_{k}$ for integers $k \ge1$. Let $b_{k} := (h_{k+1}, \operatorname{id}_{X_{k+1}}) \circ a_{k} : X_{k} \to Y_{k+1} \times X_{k+1}$, which generates the next observation $Y_{k+1}$ and hidden state $X_{k+1}$ from $X_{k}$. The joint law $\pi_{k}$ of the observations $Y^{[k]} := Y_{1} \times\cdots\times Y_{k}$ and the hidden state $X_{k}$ is the probability measure on $Y^{[k]} \times X_{k}$ given by push-forwards

of measures, and the posterior is the conditional law

giving a family of probability measures on $X_{k}$ parameterized by the observations $Y^{[k]}$ up to time $k$. The following two rules of the calculus give the Bayesian filter. Push-forward commutes with conditioning, so that

and the tower property then gives

Since $b_k\circ\pi_k(X_k\mid Y^{[k]})=(h_{k+1},\operatorname{id}_{X_{k+1}})\circ\mu_{k+1}$ for the predicted prior $\mu_{k+1}:=a_k\circ\pi_k(X_k\mid Y^{[k]}):Y^{[k]}\to X_{k+1}$, the right hand side is the Bayesian inverse $h_{k+1}^{\mu_{k+1}}$ of $h_{k+1}$ with respect to $\mu_{k+1}$, a parameterized version of the Bayesian inverse of Cho and Jacobs [5]. This gives the Bayesian filter

which states that the posterior, a priori involving the whole model up to time $k+1$, is the inverse of a single observation $h_{k+1}$, the past entering only through the predicted prior $\mu_{k+1}$.

The parameter space enables the iteration. The posterior $\pi_k(X_k\mid Y^{[k]})$ is a stochastic morphism with parameter space $Y^{[k]}$, and serves as the prior for the next observation. It also accommodates feedback. A transition or observation depending on the past observations is a stochastic morphism with parameter space $Y^{[k]}$, so that a model with feedback is simply a model whose parameter space grows with the observations. Our main results, Theorems 4.6 and 4.7, establish the Bayesian filter for the joint law $\pi$ of the model with feedback, together with the corresponding recursions for prediction and smoothing, every step being a Bayesian inversion with the past observations entering as the parameter. In the case without feedback, the filter and the smoother are those of [11], obtained in Markov categories with conditionals, where the past observations are carried as an input of the conditional.

The hidden states are standard Borel, the observation spaces only countably generated, and the parameter space arbitrary. Since we work directly with conditional laws rather than conditional expectations, no integrability is assumed, and the results are equalities of stochastic morphisms holding for every value of the parameter $s\in S$. In the affine Gaussian case with feedback, the filter and the smoother are classical for conditionally Gaussian sequences [18 Chapter 13], under integrability assumptions on the coefficients, and the filter is known to require only that the coefficients be almost surely finite [6]. In this article, the filter and the smoother follow from Theorems 4.6 and 4.7 for arbitrary measurable coefficients, in a family of affine Gaussian stochastic morphisms strictly larger than the Markov category Gauss of [10 §6]. Feedback is also natural in continuous time filtering, which will be treated in a subsequent article.

In classical terms, the conditional expectation of a bounded measurable function $f$ on $Y$ given $X$ is the pull-back

and the tower property of conditional expectations follows from that of conditional laws (Proposition B.6). In this sense, the calculus of conditional laws gives the calculus of conditional expectations.

The article is organized as follows. In §2 we first introduce conditional laws and augmentations. We then prove fundamental results such as the tower property, and formulate conditional independence. In §3 we define the Bayesian inverse and prove Bayesian coherence: updating observations successively gives the same posterior as updating them at once. In §4 we treat filtering, prediction and smoothing for hidden Markov models with feedback, and in §5 the Kalman filter and the Rauch–Tung–Striebel smoother. Appendix A gives a self-contained proof of the existence and uniqueness of the conditional law, which may be read independently, Appendix B relates our framework to classical expectations and conditional expectations, and Appendix C gives a dictionary between classical notation, the notation of Markov categories and our article.

Conditional Laws and Augmentations

Throughout this article, $S$, $X$, $Y$, $Z$, $V$, $W$ denote measurable spaces. In this section, we define the conditional law of a parameterized joint law and prove fundamental properties such as the tower property, using augmentations of stochastic morphisms.

A measurable space $X$ is countably generated if its $\sigma$-algebra is generated by a countable family of measurable subsets, and standard Borel if it is isomorphic as a measurable space to a Borel subset of a Polish space with its Borel $\sigma$-algebra, or equivalently to a Borel subset of $[0,1]$ (see [24 Theorem 3.3.13]). A standard Borel space is countably generated, and finite or countable products of standard Borel spaces are again standard Borel spaces.

A stochastic morphism $p : X \to Y$, also known as a Markov kernel, probabilistic mapping or transition probability from $X$ to $Y$, is a map sending any $x \in X$ to a probability measure $p(x)$ on $Y$, such that for any measurable $B \subset Y$, the map $x \mapsto p(x)(B)$ is measurable in $x$. Any measurable map $f : X \to Y$ can be viewed as a stochastic morphism by mapping $x$ to the Dirac measure $\delta_{f(x)}$ on $Y$. We will call such a stochastic morphism a deterministic morphism. We denote by $1 := \{\ast\}$ the one-point measurable space. A probability measure $\mu$ on $X$ defines a stochastic morphism $\mu: 1 \to X$ mapping $\ast$ to $\mu$, and we will identify the two.

Given stochastic morphisms $p : S \to X$ and $q : X \to Y$, we define the composition by

for any $s \in S$ and measurable set $B \subset Y$. Then $q \circ p$ is again a stochastic morphism. Measurable spaces and stochastic morphisms form a category, the stochastic category, denoted Stoch in [10], with the deterministic morphisms induced by the identities as identities.

We denote by $\operatorname{pr}_{X} : X \times Y \to X$ and $\operatorname{pr}_{Y} : X \times Y \to Y$ the deterministic morphisms induced by the projections. More generally, we take the convention that on a product with several factors including $X$ and $Y$, we denote by $\operatorname{pr}_{X \times Y}$ and $\operatorname{pr}_{Y \times X}$ the projections to $X \times Y$ and to $Y \times X$, the order of the indices indicating the order of the factors in the range, and similarly for projections to more than two factors. For stochastic morphisms $p : S \to X$ and $q : S \to Y$, the product