Dirichlet Filtrations: Synthetic Bayesian Modeling

• Dirichlet filtrations are a synthetic framework that replaces classical linear time with a category of context-dependent time using Dirichlet measures on simplices.
• They combine categorical probability, simplicial geometry, and Bayesian statistics to enable functorial probability transport and systematic Bayesian updating.
• This approach synthesizes multiple pasts into a unique present while introducing controlled uncertainty in future parameters, with applications in hierarchical and categorical learning.

Searching arXiv for the specified paper to ground the article with the cited source. Dirichlet filtrations are a concrete instance of the synthetic filtration framework introduced in "A geometric model of synthetic filtrations via context-dependent time" (Adachi, 16 Sep 2025). They replace linear time by a category of context-dependent time, allowing “the present” to be formed by synthesizing multiple possible pasts, and they realize this structure probabilistically by assigning Dirichlet measures on simplices to context-indexed time objects. In this formulation, categorical probability, simplicial geometry, and Bayesian statistics are combined so that probability transport along face and degeneracy maps is functorial, and Bayesian updating appears as a categorical transformation of a Dirichlet functor (Adachi, 16 Sep 2025).

1. Category of context-dependent time

The construction begins from the simplex category $\Delta$, the full subcategory of $\mathrm{Ord}$ with objects $[n] = \{0,1,\ldots,n\}$ and morphisms the order-preserving maps (Adachi, 16 Sep 2025). Two generating families of morphisms are used throughout. The face maps $\delta^n_i:[n-1]\to[n]$ insert one new value at $i$, and the degeneracy maps $\sigma^n_j:[n+1]\to[n]$ identify $j$ and $j+1$. Every morphism in $\Delta$ can be written as a composite of $\delta$’s and $\mathrm{Ord}$0’s, and these generators satisfy the standard simplicial identities (Adachi, 16 Sep 2025).

The category $\mathrm{Ord}$1 extends $\mathrm{Ord}$2 by incorporating context into the very notion of time. A context is a sequence $\mathrm{Ord}$3 with $\mathrm{Ord}$4, and the set of all contexts is

$\mathrm{Ord}$5

For $\mathrm{Ord}$6 and $\mathrm{Ord}$7, the equivalence relation

$\mathrm{Ord}$8

identifies contexts that agree from time $\mathrm{Ord}$9 onward (Adachi, 16 Sep 2025). An object of $[n] = \{0,1,\ldots,n\}$0 is a time-in-context

$[n] = \{0,1,\ldots,n\}$1

that is, time $[n] = \{0,1,\ldots,n\}$2 together with an equivalence class of contexts “from $[n] = \{0,1,\ldots,n\}$3 onward” (Adachi, 16 Sep 2025). Distinct complete contexts that agree from $[n] = \{0,1,\ldots,n\}$4 onward determine the same present object. This encodes the statement that the present may synthesize different earlier histories.

Morphisms in $[n] = \{0,1,\ldots,n\}$5 are inherited from $[n] = \{0,1,\ldots,n\}$6 but parameterized by context: for fixed $[n] = \{0,1,\ldots,n\}$7,

$[n] = \{0,1,\ldots,n\}$8

The context-indexed face and degeneracy morphisms are

$[n] = \{0,1,\ldots,n\}$9

A forgetful functor $\delta^n_i:[n-1]\to[n]$0 sends $\delta^n_i:[n-1]\to[n]$1 and sends $\delta^n_i:[n-1]\to[n]$2 to $\delta^n_i:[n-1]\to[n]$3 in $\delta^n_i:[n-1]\to[n]$4 (Adachi, 16 Sep 2025).

Intuitively, $\delta^n_i:[n-1]\to[n]$5 encodes branching and merging histories by letting the context choose, at each time, which face or degeneracy operation is used. Non-linearity comes from the fact that different contexts can lead to the same present, and non-uniqueness of the past is captured by many contexts mapping into a single object $\delta^n_i:[n-1]\to[n]$6 once one forgets pre-$\delta^n_i:[n-1]\to[n]$7 details (Adachi, 16 Sep 2025).

2. Synthetic filtrations as contravariant probabilistic functors

A synthetic filtration is a contravariant functor

$\delta^n_i:[n-1]\to[n]$8

where $\delta^n_i:[n-1]\to[n]$9 is the category of probability spaces with null-preserving maps (Adachi, 16 Sep 2025). An object of $i$0 is a probability space $i$1. A measurable map

$i$2

is null-preserving if the pushforward measure satisfies absolute continuity,

$i$3

Equivalently, sets that are $i$4-null have $i$5-pullbacks that are $i$6-null (Adachi, 16 Sep 2025).

This choice ensures well-defined conditional expectations along arrows. The formal statement used in the exposition is that for $i$7 null-preserving and integrable $i$8, there is $i$9 on $\sigma^n_j:[n+1]\to[n]$0 with

$\sigma^n_j:[n+1]\to[n]$1

The tower property also holds, so probabilistic transport along composable arrows is categorically coherent (Adachi, 16 Sep 2025).

Contravariance reverses arrows: a morphism

$\sigma^n_j:[n+1]\to[n]$2

in $\sigma^n_j:[n+1]\to[n]$3 induces

$\sigma^n_j:[n+1]\to[n]$4

Geometrically, one pushes measures forward along the underlying map in $\sigma^n_j:[n+1]\to[n]$5 and pulls random variables back. For a measurable $\sigma^n_j:[n+1]\to[n]$6, pushforward is

$\sigma^n_j:[n+1]\to[n]$7

In the Dirichlet construction, the arrow maps are either measure-preserving or null-preserving due to the Dirichlet structure (Adachi, 16 Sep 2025).

This framework generalizes the classical filtration $\sigma^n_j:[n+1]\to[n]$8 indexed by linear time. In the classical case, one may view the filtration as a contravariant functor $\sigma^n_j:[n+1]\to[n]$9 sending $j$0 and $j$1 to the identity map $j$2 (Adachi, 16 Sep 2025). The synthetic framework removes the assumption of a unique past path and replaces linear time by simplicial and context-dependent indexing.

3. Dirichlet measures on simplices and the definition of Dirichlet filtrations

Dirichlet filtrations instantiate the synthetic filtration by assigning to each object $j$3 a Dirichlet measure on a geometric simplex (Adachi, 16 Sep 2025). The geometric simplex of dimension $j$4 is

$j$5

For parameters $j$6 with $j$7, the Dirichlet density is

$j$8

with multivariate beta

$j$9

If $j+1$0, then the standard moment formulas are

$j+1$1

$j+1$2

and

$j+1$3

(Adachi, 16 Sep 2025).

The exposition also allows coordinates with $j+1$4, in which case the measure is degenerate at zero on those coordinates. If

$j+1$5

and $j+1$6 is the increasing embedding of nonzero indices, then the induced probability measure $j+1$7 on $j+1$8 is defined by restricting to the face where $j+1$9 whenever $\Delta$0 and using the Dirichlet density on the remaining coordinates:

$\Delta$1

where

$\Delta$2

and

$\Delta$3

This definition supplies a uniform treatment of ordinary and degenerate Dirichlet measures (Adachi, 16 Sep 2025).

A Dirichlet filtration is most conveniently defined through an intermediate category $\Delta$4 whose objects are pairs $\Delta$5 with $\Delta$6, where some $\Delta$7 may be zero, and whose morphisms are the corresponding null-preserving maps between Dirichlet probability spaces

$\Delta$8

(Adachi, 16 Sep 2025). A Dirichlet functor is a functor

$\Delta$9

subject to the rule “past parameters are faces of present parameters”: if

$\delta$0

then

$\delta$1

Composing with the natural functor $\delta$2 yields a $\delta$3-filtration

$\delta$4

(Adachi, 16 Sep 2025).

4. Geometric realization and measure transport

The morphisms in $\delta$5 act on Dirichlet measures through the geometric realizer’s face and degeneracy maps (Adachi, 16 Sep 2025). The contravariant geometric realizer $\delta$6 supplies continuous maps on simplices. The face map

$\delta$7

“merges” coordinates $\delta$8 and $\delta$9. For

$\mathrm{Ord}$00

it is given by

$\mathrm{Ord}$01

The degeneracy map

$\mathrm{Ord}$02

“inserts” a zero at position $\mathrm{Ord}$03:

$\mathrm{Ord}$04

These maps satisfy the simplicial identities (Adachi, 16 Sep 2025).

The central transport identities are

$\mathrm{Ord}$05

The first identity is the Dirichlet reproduction property: summing adjacent coordinates yields a lower-dimensional Dirichlet with parameters summed accordingly. The second encodes the insertion of a zero coordinate at $\mathrm{Ord}$06 at the measure level together with the corresponding parameter transformation (Adachi, 16 Sep 2025).

These transport laws are the backbone of functoriality for Dirichlet filtrations. The geometric face and degeneracy maps act compatibly on both space and parameters, and in the paper’s precise sense they are measure-preserving (Adachi, 16 Sep 2025). Because $\mathrm{Ord}$07 and $\mathrm{Ord}$08 are continuous, measurability follows from the Borel $\mathrm{Ord}$09-algebras $\mathrm{Ord}$10, and because the pushforward identities hold exactly, the required absolute continuity for null-preservation holds trivially.

A plausible implication is that the formal role of simplicial geometry here is not merely organizational: it directly determines how probabilistic state spaces are merged or extended. In particular, merging coordinates is not an external aggregation rule but the categorical image of a face map.

5. Bayesian updating and categorical learning

Bayesian learning events are implemented by natural transformations of Dirichlet functors (Adachi, 16 Sep 2025). If at time-in-context $\mathrm{Ord}$11 one observes multinomial data with counts

$\mathrm{Ord}$12

then conjugacy gives the posterior Dirichlet at that object:

$\mathrm{Ord}$13

This update is localized at $\mathrm{Ord}$14, but functoriality determines how it propagates across the filtration.

The updated past is uniquely determined. For any $\mathrm{Ord}$15 and context $\mathrm{Ord}$16 with $\mathrm{Ord}$17,

$\mathrm{Ord}$18

with

$\mathrm{Ord}$19

This is the formal statement that past values are obtained by iterated faces applied to the posterior parameter (Adachi, 16 Sep 2025).

The future behaves differently. At time $\mathrm{Ord}$20, with $\mathrm{Ord}$21,

$\mathrm{Ord}$22

may be any $\mathrm{Ord}$23 satisfying

$\mathrm{Ord}$24

Equivalently,

$\mathrm{Ord}$25

so there are degrees of freedom in how the newly inserted zero coordinate at position $\mathrm{Ord}$26 is allocated by subsequent learning or modeling choices (Adachi, 16 Sep 2025). The paper describes this as the categorical update law and as learning expressed by a natural transformation

$\mathrm{Ord}$27

that propagates backward strictly functorially and forward with constrained freedom.

This asymmetry between past and future is structurally significant. Past values are fixed by face maps, while future values retain parameter uncertainty constrained by context. The exposition explicitly identifies both parameter uncertainty and contextual uncertainty: the first concerns the choice of $\mathrm{Ord}$28 satisfying the face constraint, and the second concerns which face or degeneracy will be used next (Adachi, 16 Sep 2025).

6. Synthesis of multiple pasts, classical reduction, example, and limitations

The present object $\mathrm{Ord}$29 depends only on the equivalence class $\mathrm{Ord}$30, so distinct contexts that agree from $\mathrm{Ord}$31 onward synthesize into the same present (Adachi, 16 Sep 2025). Categorically, this arises as the quotient by $\mathrm{Ord}$32 on $\mathrm{Ord}$33 and can be seen as a coequalization of the earlier branching maps: many arrows from different pasts feed into a common object via the same face map $\mathrm{Ord}$34 (Adachi, 16 Sep 2025). The geometric realizer translates synthesis into the merging of coordinates in simplices by $\mathrm{Ord}$35.

Dirichlet priors naturally accommodate this synthesis. If multiple contexts lead to the same present parameters $\mathrm{Ord}$36 on $\mathrm{Ord}$37, then the present Dirichlet measure $\mathrm{Ord}$38 represents uncertainty about future outcomes independently of which precise past path occurred (Adachi, 16 Sep 2025). The exposition further states that parameter uncertainty and contextual uncertainty can be superposed, and that mixtures over contexts or hierarchies, for example hyperpriors on $\mathrm{Ord}$39, fit seamlessly by viewing $\mathrm{Ord}$40-indexed families of $\mathrm{Ord}$41’s and integrating over context distributions (Adachi, 16 Sep 2025). This suggests a route toward hierarchical and mixture constructions within the same indexing scheme.

The framework reduces to the classical case under a restriction. If one fixes a single context $\mathrm{Ord}$42 and restricts to the subcategory generated by face maps that deterministically select the same index, for example $\mathrm{Ord}$43, together with identity maps on $\mathrm{Ord}$44, then $\mathrm{Ord}$45 collapses to a chain isomorphic to $\mathrm{Ord}$46 and the filtration behaves classically (Adachi, 16 Sep 2025). This reduction clarifies that the novelty of Dirichlet filtrations is not the use of probability spaces alone, but the replacement of unique linear history by simplicial, context-controlled branching and merging.

A concrete worked example is given for $\mathrm{Ord}$47 (Adachi, 16 Sep 2025). Let $\mathrm{Ord}$48, let the state space be

$\mathrm{Ord}$49

and let the prior be the uniform Dirichlet

$\mathrm{Ord}$50

After observing a single multinomial outcome “category 1,” with counts

$\mathrm{Ord}$51

conjugacy yields

$\mathrm{Ord}$52

and posterior expectations

$\mathrm{Ord}$53

For the past time $\mathrm{Ord}$54, with $\mathrm{Ord}$55, applying the face map $\mathrm{Ord}$56 to parameters merges coordinates $\mathrm{Ord}$57 and $\mathrm{Ord}$58:

$\mathrm{Ord}$59

so

$\mathrm{Ord}$60

a Beta distribution on $\mathrm{Ord}$61 with mean

$\mathrm{Ord}$62

For the future at $\mathrm{Ord}$63, if $\mathrm{Ord}$64, then $\mathrm{Ord}$65 must satisfy

$\mathrm{Ord}$66

If $\mathrm{Ord}$67, this imposes

$\mathrm{Ord}$68

leaving a one-dimensional degree of freedom in how the mass is split between coordinates $\mathrm{Ord}$69 and $\mathrm{Ord}$70 (Adachi, 16 Sep 2025).

The main limitations identified in the exposition are likewise structural. The construction is discrete-time and simplicial; extending it to continuous time would require appropriate geometric structures and a continuous analogue of context-dependent faces and degeneracies (Adachi, 16 Sep 2025). Degeneracy maps insert zero coordinates and are non-invertible, so future parameter uncertainty is intrinsic and must be managed by context or additional modeling, for example splitting rules (Adachi, 16 Sep 2025). The development focuses on parametric Dirichlet priors, while nonparametric generalizations, other conjugate families, and hierarchical models indexed by $\mathrm{Ord}$71 are described as natural extensions. Homological or homotopical invariants of $\mathrm{Ord}$72-filtrations and applications in finance and economics with non-linear information flow are presented as open directions (Adachi, 16 Sep 2025).