An Operadic View of the Law of Total Probability

An Operadic View of the Law of Total Probability1. The Operad of Probability Simplices2. What Does the Composition Mean?3. Measurable Partitions as a Colored Operad 4. Probability as an Operad Morphism5. The Simplex Operad Acts on $[0,1]$6. The Law of Total Probability7. The Law of Total Probability as a Morphism of Algebras7.1 The Terminal Algebra7.2 A Fixed Event Determines a Family of Elements7.3 The Algebra-Morphism Condition7.4 What Does This Reformulation Add?8. Further Refinement9. The Basic DiagramAppendix A. Colored OperadsAppendix B. Morphisms of Colored OperadsAppendix C. Algebras and Algebra MorphismsReferencesCollaboration Report

 

An Operadic View of the Law of Total Probability

The standard simplex

$${\mathrm{\Delta }}^{n-1}=\{(p_1,\dots ,p_n)\in {\mathbb{R}}_{\ge 0}^n|\sum _{i=1}^n{p}_i=1\}$$

may be identified with the space of probability measures on an $n$$n$-element set. Under this interpretation, the standard simplices carry a natural operadic structure: operadic composition corresponds to refining one branch of a probability tree.

This simple observation leads to an operadic interpretation of the law of total probability.

The basic notions of symmetric colored operads, their morphisms, algebras, and algebra morphisms are recalled in the appendix. We use the terms symmetric colored operad and symmetric multicategory interchangeably; see Leinster [Lei04].

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1. The Operad of Probability Simplices

For $n\ge 1$$n\ge 1$, define

$$\mathcal{P}(n):={\mathrm{\Delta }}^{n-1}.$$

Thus an element

$$p=(p_1,\dots ,p_n)\in \mathcal{P}(n)$$

is an $n$$n$-tuple of nonnegative numbers satisfying

$$\sum _{i=1}^n{p}_i=1.$$

We may regard $p$$p$ either as a point of the standard simplex or as a probability distribution on the finite set

$$\{1,\dots ,n\}.$$

The operadic structure is most transparent in terms of partial composition.

Let

$$p=(p_1,\dots ,p_n)\in \mathcal{P}(n)$$

and

$$q=(q_1,\dots ,q_m)\in \mathcal{P}(m).$$

For $1\le i\le n$$1\le i\le n$, define

$$\boxed{{\displaystyle p{\circ }_{i}q=(p_1,\dots ,p_{i-1},p_i{q}_1,\dots ,p_i{q}_m,p_{i+1},\dots ,p_n).}}$$

Thus the $i$$i$-th probability mass $p_i$$p_i$ is replaced by $m$$m$ smaller masses

$$p_i{q}_1,\dots ,p_i{q}_m.$$

Since

$$\sum _{k\ne i}{p}_k+p_i\sum _{j=1}^m{q}_j=\sum _{k\ne i}{p}_k+p_i=1,$$

we indeed obtain an element of

$$\mathcal{P}(n+m-1).$$

The unit is

$$(1)\in \mathcal{P}(1)={\mathrm{\Delta }}^0,$$

and the symmetric group acts by permuting coordinates.

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2. What Does the Composition Mean?

Consider

$$p=(\frac{1}{2},\frac{1}{3},\frac{1}{6})\in \mathcal{P}(3).$$

Interpret this as a random choice among three alternatives

$$A_1,A_2,A_3$$

with probabilities

$$P(A_1)=\frac{1}{2},P(A_2)=\frac{1}{3},P(A_3)=\frac{1}{6}.$$

Suppose that we now refine only the second alternative. Conditional on being in $A_2$$A_2$, let there be two further alternatives

$$B_1,B_2$$

with conditional distribution

$$q=(\frac{1}{4},\frac{3}{4})\in \mathcal{P}(2).$$

Then

$$p{\circ }_{2}q=(\frac{1}{2},\frac{1}{3}\frac{1}{4},\frac{1}{3}\frac{3}{4},\frac{1}{6}),$$

hence

$$\boxed{{\displaystyle p{\circ }_{2}q=(\frac{1}{2},\frac{1}{12},\frac{1}{4},\frac{1}{6}).}}$$

The four final branches have probabilities

$$\frac{1}{2},\frac{1}{12},\frac{1}{4},\frac{1}{6}.$$

Thus the meaning of ${\circ }_i$$\circ_i$ is:

$$\boxed{{\displaystyle p{\circ }_{i}q=\text{refine the }i\text{-th branch according to the conditional distribution }q\text{.}}}$$

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3. Measurable Partitions as a Colored Operad

Now fix a probability space

$$(\mathrm{\Omega },\mathrm{\Sigma },P).$$

For the moment, we restrict to measurable events $A$$A$ satisfying

$$P(A)>0,$$

and to partitions whose pieces also have positive probability. This avoids the ambiguity of conditioning on null events.

Define a colored operad ${\mathcal{D}}_P$$\mathcal D_P$ as follows.

Its colors are the positive-probability measurable events

$$A\in \mathrm{\Sigma }.$$

There is an operation

$$(A_1,\dots ,A_n)⟶A$$

precisely when

$$\boxed{{\displaystyle A=A_1\bigsqcup \cdots \bigsqcup A_n.}}$$

Thus an operation is a finite ordered measurable partition of $A$$A$.

The colored structure records exactly which event is being refined. If

$$A=A_1\bigsqcup \cdots \bigsqcup A_n$$

and

$$A_i=B_1\bigsqcup \cdots \bigsqcup B_m,$$

then the second decomposition may be inserted only into the $i$$i$-th input slot of the first.

Their partial composition is the refined partition

$$\boxed{{\displaystyle (A_1,\dots ,A_n){\circ }_i(B_1,\dots ,B_m)=(A_1,\dots ,A_{i-1},B_1,\dots ,B_m,A_{i+1},\dots ,A_n).}}$$

The output color remains $A$$A$.

Thus

$$\boxed{{\displaystyle \text{operadic composition in }{\mathcal{D}}_P=\text{refinement of measurable partitions}.}}$$

Associativity says that successive refinements may be grouped in any order without changing the final partition.

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4. Probability as an Operad Morphism

The probability measure now gives a natural map from the partition colored operad to the simplex operad.

Since $\mathcal{P}$$\mathcal P$ is an ordinary operad, we regard it as a one-colored operad. On colors there is therefore only one possible map:

$$A⟼\ast .$$

For an operation

$$A=A_1\bigsqcup \cdots \bigsqcup A_n,$$

define

$$\boxed{{\displaystyle {\mathrm{\Phi }}_P(A_1,\dots ,A_n;A)=(P(A_1\mid A),\dots ,P(A_n\mid A))\in {\mathrm{\Delta }}^{n-1}.}}$$

We claim that

$$\boxed{{\displaystyle {\mathrm{\Phi }}_P:{\mathcal{D}}_P⟶\mathcal{P}}}$$

is a morphism of colored operads.

The essential point is compatibility with partial composition.

Suppose

$$A=A_1\bigsqcup \cdots \bigsqcup A_n$$

and the $i$$i$-th piece is further decomposed as

$$A_i=B_1\bigsqcup \cdots \bigsqcup B_m.$$

Write

$$p_k=P(A_k\mid A)$$

and

$$q_j=P(B_j\mid A_i).$$

The partial composite in the simplex operad is

$${\mathrm{\Phi }}_P(A_1,\dots ,A_n;A){\circ }_i{\mathrm{\Phi }}_P(B_1,\dots ,B_m;A_i),$$

whose new coordinates in the $i$$i$-th slot are

$$p_i{q}_j=P(A_i\mid A)P(B_j\mid A_i).$$

Since $B_j\subseteq A_i\subseteq A$$B_j\subseteq A_i\subseteq A$,

$$\begin{array}{rl}P(A_i\mid A)P(B_j\mid A_i)& =\frac{P(A_i)}{P(A)}\frac{P(B_j)}{P(A_i)}\\ & =\frac{P(B_j)}{P(A)}\\ & =P(B_j\mid A).\end{array}$$

Hence

$$\boxed{{\displaystyle {\mathrm{\Phi }}_P(\theta {\circ }_i\varphi )={\mathrm{\Phi }}_P(\theta ){\circ }_i{\mathrm{\Phi }}_P(\varphi ).}}$$

So the elementary identity

$$\boxed{{\displaystyle P(B_j\mid A)=P(A_i\mid A)P(B_j\mid A_i)}}$$

is exactly the condition that probability preserve operadic composition.

This gives a structural interpretation of conditional probability:

$$\boxed{{\displaystyle \text{measurable refinement}\stackrel{{\mathrm{\Phi }}_P}{⟼}\text{probability-tree refinement}.}}$$

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5. The Simplex Operad Acts on $[0,1]$$[0,1]$

The interval $[0,1]$$[0,1]$ is convex. Hence every

$$p=(p_1,\dots ,p_n)\in \mathcal{P}(n)$$

determines an $n$$n$-ary operation

$${\beta }_p:[0,1{]}^n⟶[0,1]$$

given by

$$\boxed{{\displaystyle {\beta }_p(r_1,\dots ,r_n)=\sum _{i=1}^n{p}_i{r}_i.}}$$

These operations are compatible with operadic composition. Indeed,

$$\begin{array}{rl}{\beta }_p(r_1,\dots ,{\beta }_q(s_1,\dots ,s_m),\dots ,r_n)& =\sum _{k\ne i}{p}_k{r}_k+p_i\sum _j{q}_j{s}_j\\ & =\sum _{k\ne i}{p}_k{r}_k+\sum _j{p}_i{q}_j{s}_j,\end{array}$$

which is precisely the operation corresponding to

$$p{\circ }_{i}q.$$

Thus $[0,1]$$[0,1]$ is naturally a $\mathcal{P}$$\mathcal P$-algebra.

Equivalently, in $\mathbf{Set}$$\mathbf{Set}$,

$$\boxed{{\displaystyle \alpha :\mathcal{P}⟶{\mathrm{End}}_{[0,1]}}}$$

is an operad morphism.

Combining this with ${\mathrm{\Phi }}_P$$\Phi_P$, we obtain

$$\boxed{{\displaystyle {\mathcal{D}}_P\stackrel{{\mathrm{\Phi }}_P}{\to }\mathcal{P}\stackrel{\alpha }{\to }{\mathrm{End}}_{[0,1]}.}}$$

Consequently, every measurable partition

$$A=A_1\bigsqcup \cdots \bigsqcup A_n$$

acts on $[0,1{]}^n$$[0,1]^n$ by

$$\boxed{{\displaystyle (r_1,\dots ,r_n)⟼\sum _{i}P(A_i\mid A)r_i.}}$$

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6. The Law of Total Probability

Fix an event

$$B\in \mathrm{\Sigma }.$$

For each positive-probability event $A$$A$, define

$$s_B(A):=P(B\mid A)\in [0,1].$$

Now let

$$A=A_1\bigsqcup \cdots \bigsqcup A_n.$$

The operation associated to this partition is

$$(r_1,\dots ,r_n)⟼\sum _{i}P(A_i\mid A)r_i.$$

Substituting

$$r_i=P(B\mid A_i)$$

gives

$$\sum _{i}P(A_i\mid A)P(B\mid A_i).$$

The law of total probability states that this is exactly

$$P(B\mid A).$$

Hence

$$\boxed{{\displaystyle s_B(A)={\beta }_{{\mathrm{\Phi }}_P(A_1,\dots ,A_n;A)}(s_B(A_1),\dots ,s_B(A_n)).}}$$

Equivalently,

$$\boxed{{\displaystyle P(B\mid A)=\sum _{i}P(A_i\mid A)P(B\mid A_i).}}$$

Thus the law of total probability says that the family

$$A⟼P(B\mid A)$$

is compatible with the operations of the partition colored operad.

When $A=\mathrm{\Omega }$$A=\Omega$, this reduces to the usual formula

$$\boxed{{\displaystyle P(B)=\sum _{i}P(A_i)P(B\mid A_i).}}$$

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7. The Law of Total Probability as a Morphism of Algebras

The preceding interpretation can be sharpened further.

Recall that the composite

$${\mathcal{D}}_P\stackrel{{\mathrm{\Phi }}_P}{\to }\mathcal{P}\stackrel{\alpha }{\to }{\mathrm{End}}_{[0,1]}$$

defines a ${\mathcal{D}}_P$$\mathcal D_P$-algebra $X$$X$ by assigning

$$X_A=[0,1]$$

to every color $A$$A$.

For each measurable partition

$$\theta :(A_1,\dots ,A_n)⟶A,A=\underset{i=1}{\overset{n}{⨆}}{A}_i,$$

the corresponding algebra operation is

$${\mu }_{\theta }:X_{A_1}\times \cdots \times X_{A_n}⟶X_A,$$

given explicitly by

$$\boxed{{\displaystyle {\mu }_{\theta }(r_1,\dots ,r_n)=\sum _{i=1}^{n}P(A_i\mid A)r_i.}}$$

Thus the colored operad ${\mathcal{D}}_P$$\mathcal D_P$ acts on the family of intervals

$$(X_A{)}_A$$

through conditional-probability-weighted barycentric combinations.

The law of total probability can now be expressed as a morphism into this algebra.

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7.1 The Terminal Algebra

For every color $A$$A$, let

$${\mathbf{1}}_A=\{\ast \}$$

be the one-point set.

These one-point sets form a ${\mathcal{D}}_P$$\mathcal D_P$-algebra $\mathbf{1}$$\mathbf 1$.

Indeed, for every operation

$$\theta :(A_1,\dots ,A_n)\to A,$$

there is exactly one possible map

$${\mathbf{1}}_{A_1}\times \cdots \times {\mathbf{1}}_{A_n}⟶{\mathbf{1}}_A.$$

Explicitly,

$$(\ast ,\dots ,\ast )⟼\ast .$$

All operadic compatibility conditions are automatic, since every relevant map between singleton sets is unique.

Moreover, $\mathbf{1}$$\mathbf 1$ is the terminal object in the category of ${\mathcal{D}}_P$$\mathcal D_P$-algebras: for every ${\mathcal{D}}_P$$\mathcal D_P$-algebra $Y$$Y$, there is a unique algebra morphism

$$Y⟶\mathbf{1}.$$

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7.2 A Fixed Event Determines a Family of Elements

Now fix an event

$$B\in \mathrm{\Sigma }.$$

For every positive-probability event $A$$A$, we have an element

$$P(B\mid A)\in X_A=[0,1].$$

Equivalently, an element of $X_A$$X_A$ may be regarded as a map from a singleton:

$$s_{B,A}:{\mathbf{1}}_A⟶X_A,$$

defined by

$$\boxed{{\displaystyle s_{B,A}(\ast )=P(B\mid A).}}$$

Thus a fixed event $B$$B$ determines a family of maps

$$s_B=(s_{B,A}{)}_A:\mathbf{1}⟶X.$$

At this point, however, $s_B$$s_B$ is merely a family of maps indexed by the colors.

The question is:

When is this family a morphism of ${\mathcal{D}}_P$$\mathcal D_P$-algebras?

The answer is precisely: when the law of total probability holds.

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7.3 The Algebra-Morphism Condition

By definition, $s_B:\mathbf{1}\to X$$s_B:\mathbf 1\to X$ is a morphism of ${\mathcal{D}}_P$$\mathcal D_P$-algebras if, for every operation

$$\theta :(A_1,\dots ,A_n)⟶A,$$

the diagram

$$\begin{array}{ccc}{\mathbf{1}}_{A_1}\times \cdots \times {\mathbf{1}}_{A_n}& \stackrel{{\mu }_{\theta }^{\mathbf{1}}}{\to }& {\mathbf{1}}_A\\ s_{B,A_1}\times \cdots \times s_{B,A_n}\downarrow & & \downarrow s_{B,A}& \\ X_{A_1}\times \cdots \times X_{A_n}& \stackrel{{\mu }_{\theta }^X}{\to }& X_A\end{array}$$

commutes.

Since the upper row is the unique map between singleton sets, there is only one input to check:

$$(\ast ,\dots ,\ast ).$$

Going first across the top and then down the right gives

$$\ast ⟼\ast ⟼P(B\mid A).$$

Going first down the left and then across the bottom gives

$$(\ast ,\dots ,\ast )⟼(P(B\mid A_1),\dots ,P(B\mid A_n))$$

and then

$$\begin{array}{rl}{\mu }_{\theta }^X(P(B\mid A_1),\dots ,P(B\mid A_n))& =\sum _{i}P(A_i\mid A)P(B\mid A_i).\end{array}$$

Therefore the diagram commutes if and only if

$$\boxed{{\displaystyle P(B\mid A)=\sum _{i}P(A_i\mid A)P(B\mid A_i).}}$$

But this is exactly the conditional form of the law of total probability.

Hence:

$$\boxed{{\displaystyle s_B:\mathbf{1}⟶X\text{ is a }{\mathcal{D}}_P\text{-algebra morphism}}}$$

if and only if

$$\boxed{{\displaystyle P(B\mid A)=\sum _{i}P(A_i\mid A)P(B\mid A_i)}}$$

for every admissible measurable partition

$$A=\underset{i}{⨆}{A}_i.$$

Thus the law of total probability can be reformulated as the statement

$$\boxed{{\displaystyle \text{For every fixed event }B,\text{ the family }A\mapsto P(B\mid A)\text{ is a compatible point of the }{\mathcal{D}}_P\text{-algebra }X.}}$$

Equivalently, this compatible point is the algebra morphism

$$\boxed{{\displaystyle s_B:\mathbf{1}\to X.}}$$

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7.4 What Does This Reformulation Add?

At first sight, this may appear to be only a categorical rewriting of the familiar identity

$$P(B\mid A)=\sum _{i}P(A_i\mid A)P(B\mid A_i).$$

However, the algebra-morphism formulation packages a stronger structural statement.

The law of total probability is not attached to a single partition. For fixed $B$$B$, it holds simultaneously for every operation of the colored operad ${\mathcal{D}}_P$$\mathcal D_P$:

$$(A_1,\dots ,A_n)\to A.$$

Thus the family

$$\{P(B\mid A){\}}_A$$

is compatible with the entire system of measurable refinements.

In particular, if

$$A=\underset{i}{⨆}{A}_i$$

and each $A_i$$A_i$ is further refined as

$$A_i=\underset{j}{⨆}{A}_{ij},$$

then the algebra-morphism condition may be applied first to each $A_i$$A_i$ and then to $A$$A$, giving

$$P(B\mid A)=\sum _{i}P(A_i\mid A)(\sum _{j}P(A_{ij}\mid A_i)P(B\mid A_{ij})).$$

Because ${\mathrm{\Phi }}_P$$\Phi_P$ preserves operadic composition,

$$P(A_i\mid A)P(A_{ij}\mid A_i)=P(A_{ij}\mid A),$$

and therefore

$$P(B\mid A)=\sum _{i,j}P(A_{ij}\mid A)P(B\mid A_{ij}).$$

Hence the compatibility encoded by $s_B$$s_B$ is automatically stable under arbitrary operadic refinement.

We may therefore separate the structure into three levels:

$$\begin{array}{c}\boxed{{\displaystyle \begin{array}{rcl}{\mathcal{D}}_P& :& \text{how measurable events are refined},\\ {\mathrm{\Phi }}_P:{\mathcal{D}}_P\to \mathcal{P}& :& \text{how probability weights behave under refinement},\\ X& :& \text{how those weights act by barycentric combination}.\end{array}}}\end{array}$$

For a fixed event $B$$B$, the morphism

$$s_B:\mathbf{1}\to X$$

then says that the conditional probabilities

$$P(B\mid A)$$

respect this entire compositional structure.

This gives a concise operadic interpretation of the law of total probability:

$$\boxed{{\displaystyle \text{the law of total probability is the compatibility of conditional probability with measurable refinement.}}}$$

Or, in purely operadic language,

$$\boxed{{\displaystyle \text{the law of total probability is an algebra-morphism condition.}}}$$

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8. Further Refinement

Suppose

$$A=\underset{i}{⨆}{A}_i,A_i=\underset{j}{⨆}{A}_{ij}.$$

Applying the law of total probability first inside each $A_i$$A_i$ gives

$$P(B\mid A_i)=\sum _{j}P(A_{ij}\mid A_i)P(B\mid A_{ij}).$$

Applying it again at the outer level gives

$$P(B\mid A)=\sum _{i}P(A_i\mid A)(\sum _{j}P(A_{ij}\mid A_i)P(B\mid A_{ij})).$$

Using

$$P(A_i\mid A)P(A_{ij}\mid A_i)=P(A_{ij}\mid A),$$

we obtain

$$\boxed{{\displaystyle P(B\mid A)=\sum _{i,j}P(A_{ij}\mid A)P(B\mid A_{ij}).}}$$

This familiar refinement invariance is the combined effect of:

1. associativity in the partition colored operad;

2. the fact that ${\mathrm{\Phi }}_P$$\Phi_P$ preserves operadic composition;

3. the compatibility of the simplex-operad action on $[0,1]$$[0,1]$.

In other words, a probability tree may be evaluated level by level or flattened first. The result is the same.

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9. The Basic Diagram

The whole construction may be summarized by

$$\boxed{{\displaystyle {\mathcal{D}}_P\stackrel{{\mathrm{\Phi }}_P}{\to }\mathcal{P}\stackrel{\alpha }{\to }{\mathrm{End}}_{[0,1]}.}}$$

The three terms have distinct meanings:

$${\mathcal{D}}_P=\text{measurable partitions and their refinements},$$

$$\mathcal{P}(n)={\mathrm{\Delta }}^{n-1}=\text{finite probability distributions},$$

and

$$\alpha (p_1,\dots ,p_n)=[(r_1,\dots ,r_n)\mapsto \sum _i{p}_i{r}_i].$$

Thus

$$\boxed{{\displaystyle \text{partition}⟼\text{conditional probability weights}⟼\text{weighted average}.}}$$

From this point of view, the law of total probability is not an isolated summation identity. It expresses the compatibility between these three compositional structures.

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Appendix A. Colored Operads

We recall only the conventions used above. For a systematic treatment in the language of multicategories, see Leinster [Lei04].

Let $C$$C$ be a set of colors.

A symmetric $C$$C$-colored operad $\mathcal{O}$$\mathcal O$ consists of sets

$$\mathcal{O}(c_1,\dots ,c_n;c)$$

for colors $c_1,\dots ,c_n,c\in C$$c_1,\dots,c_n,c\in C$, together with units, symmetric group actions, and composition maps.

An element

$$\theta \in \mathcal{O}(c_1,\dots ,c_n;c)$$

is thought of as an operation

$$(c_1,\dots ,c_n)⟶c.$$

If

$$\theta \in \mathcal{O}(c_1,\dots ,c_n;c)$$

and

$$\varphi \in \mathcal{O}(d_1,\dots ,d_m;c_i),$$

then the partial composite

$$\theta {\circ }_i\varphi$$

is obtained by inserting $\varphi$$\phi$ into the $i$$i$-th input of $\theta$$\theta$.

The associativity axioms express the independence of iterated substitution from the order in which a rooted operation tree is composed.

An ordinary operad is precisely a one-colored colored operad.

A category may be regarded as a colored operad in which all non-unary operation sets are empty.

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Appendix B. Morphisms of Colored Operads

Let $\mathcal{O}$$\mathcal O$ and $\mathcal{Q}$$\mathcal Q$ have color sets $C$$C$ and $D$$D$.

A morphism

$$F:\mathcal{O}⟶\mathcal{Q}$$

consists of a map of colors

$$F_0:C⟶D$$

and maps

$$\mathcal{O}(c_1,\dots ,c_n;c)⟶\mathcal{Q}(F_0{c}_1,\dots ,F_0{c}_n;F_{0}c)$$

compatible with units, permutations, and operadic composition.

In particular,

$$\boxed{{\displaystyle F(\theta {\circ }_i\varphi )=F(\theta ){\circ }_{i}F(\varphi ).}}$$

The source and target operads need not have the same set of colors.

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Appendix C. Algebras and Algebra Morphisms

Let $\mathcal{O}$$\mathcal O$ be a colored operad in a symmetric monoidal category

$$(\mathcal{V},\otimes ,\mathbb{1}).$$

An $\mathcal{O}$$\mathcal O$-algebra assigns an object $X_c\in \mathcal{V}$$X_c\in\mathcal V$ to every color $c$$c$ and provides compatible action maps

$$\mathcal{O}(c_1,\dots ,c_n;c)\otimes X_{c_1}\otimes \cdots \otimes X_{c_n}⟶X_c.$$

No closedness assumption on $\mathcal{V}$$\mathcal V$ is required for this definition.

If $\mathcal{V}$$\mathcal V$ is closed symmetric monoidal, one may equivalently package the structure using the internal endomorphism operad

$${\mathrm{End}}_X(c_1,\dots ,c_n;c)=[X_{c_1}\otimes \cdots \otimes X_{c_n},X_c]$$

and a morphism

$$\mathcal{O}⟶{\mathrm{End}}_X.$$

A morphism $f:X\to Y$$f:X\to Y$ of $\mathcal{O}$$\mathcal O$-algebras is a family

$$f_c:X_c\to Y_c$$

such that every operadic operation is preserved:

$$f_c({\theta }_X(x_1,\dots ,x_n))={\theta }_Y(f_{c_1}(x_1),\dots ,f_{c_n}(x_n)).$$

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References

[Lei04] Tom Leinster, Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series 298, Cambridge University Press, 2004.

Collaboration Report

This note grew out of a sequence of discussions between Marco and ChatGPT concerning operadic structures in probability theory.

Marco's contribution. Marco initiated the main conceptual direction of the note. In particular, he observed that the standard simplex ${\mathrm{\Delta }}^{n-1}$$\Delta^{n-1}$ can be viewed as the space of probability measures on an $n$$n$-element set and proposed interpreting its operadic composition in terms of refinement of probability distributions. He further proposed organizing measurable partitions of a probability space as a colored operad and investigated how the law of total probability should be understood through operad morphisms and operad algebras. The overall mathematical motivation, the connection between refinement and conditional probability, and the decision to formulate the construction using partial compositions ${\circ }_i$$\circ_i$ were driven by Marco.

ChatGPT's contribution. ChatGPT assisted in formalizing and checking these ideas. This included making the colored-operad structure on measurable partitions precise, formulating the morphism

$${\mathrm{\Phi }}_P:{\mathcal{D}}_P\to \mathcal{P},$$

verifying that its compatibility with partial composition is exactly the identity

$$P(B_j\mid A)=P(A_i\mid A)P(B_j\mid A_i),$$

and identifying the natural action of the simplex operad on $[0,1]$$[0,1]$ by barycentric combinations. ChatGPT also helped formulate the stronger observation that, for fixed $B$$B$, the law of total probability can be expressed as the condition that

$$s_B:\mathbf{1}\to X$$

is a morphism of ${\mathcal{D}}_P$$\mathcal D_P$-algebras. It additionally assisted with exposition, notation, references, and consistency checks.

The resulting note is therefore based on Marco's mathematical questions and structural insights, with ChatGPT serving primarily as a tool for formalization, verification, and exposition.