Samuel Tenka

Problyhedra

These are some notes relating to my talk about convex spaces and probability theory.

Beyond Simplices: the Category $Conv$

The Category of Convex Spaces

Probability Simplices

Convex Spaces

Intuitively, a convex space is a space closed under convex combinations. Precisely, we consider a set $X$ equipped with a mixture operation $\mu: \Delta_X \to X$ from the space $\Delta_X \triangleq \left\{ f:X\to\RR_{\geq 0}: |\text{supp}(f)|\lt\infty\,\,\,\text{and}\,\,\,\sum_x f(x)=1 \right\}$ of formal finitary convex combinations of $X$'s elements to $X$. We say that $X$ is a convex space provided that $\mu(x \mapsto \delta_{x,y}) = y$ for each $y\in Y$ and that $\mu(\sum_{i\in I} c_i f_i) = \mu(x \mapsto \sum_{i\in I} c_i \delta_{x,\mu(f_i)})$ for all finite $I$ and all sequences of $c_i\in \RR_{\geq 0}$ and $f_i$ for which the two sides are defined. Each valid $\mu$ is determined by its restriction to functions with size-$2$ support, so we will sometimes conflate $\mu$ and this restriction, and for convenience we will sometimes write $\mu(x \mapsto \alpha \delta_{x,v} + (1-\alpha) \delta_{x,w}) \in X$ as $\alpha v + (1-\alpha) w \in X$ when $v, w \in X$.

Affine Morphisms

The morphisms in the category of convex spaces are those that preserve $\mu$. More precisely, we observe that $\Delta$ is an endofunctor on the category of Sets, and we declare the morphisms from $(X,\mu_X)$ to $(Y, \mu_Y)$ to be those functions $T:X \to Y$ such that $T \circ \mu_X = \mu_Y \circ \Delta(T)$. We have defined a category. We could have defined the category in fewer words --- as the Eilenberge Moore category for $\Delta$ --- but we choose here to be very concrete.

Exotic Spaces

Not every convex space is a subspace of some $\RR^n$. A source of examples comes from the semilattices, i.e. the posets $X$ with binary joins $\vee$: a canonical mixture operation on $X$ is determined by is determined by the rule $\alpha v + (1-\alpha) w \triangleq v \vee w$ as $v,w$ range through $X$ and $\alpha$ ranges through $(0,1)$. For example, say $X = \{\text{no}, \text{maybe}, \text{yes}\}$, where $\text{no},\text{yes}\lt\text{maybe}$. Then we have $(1/3) \text{no} + (2/3) \text{yes} = \text{maybe}$ and $(4/7) \text{no} + (3/7) \text{maybe} = \text{maybe}$. This is a space not of probabilities but of possibilities! Likewise, if $X = \{\text{zero}, \text{poly}, \text{exp}\}$ with $\text{zero}\lt\text{poly}\lt\text{exp}$, then we have $(2/5) \text{zero} + (3/5) \text{poly} = \text{poly}$ and $(5/6) \text{poly} + (1/6) \text{exp} = \text{exp}$: we have a space of sizes.

Examples of Morphism Composition

Marginalization

Expectations

Abstraction to Possibilities

Monoidal Structure and Useful Adjoints

Products and Independence

Existence and Interpretation of Products

A Pullback Example

Coproducts and Tagged Mixtures

Existence and Interpretation of Coproducts

A Pushout Example

Tensor, Hom, and Bayes

Hom and Conditional Distributions

Existence and Interpretation of Internal Homs

Hom from Square to Interval

Tensor and Joint Distributions

Existence and Interpretation of Tensor Products

Product vs Coproduct vs Tensor of Two Intervals

Bayes' Theorem

Visualizing the Monty Hall Problem

Statistics: Fisher, Basu, and Bahadur

Statistics and their Properties

What is a Statistic?

Sufficiency, Mimimality, Completeness, and Ancillarity

Some Basic Theorems

Fisher's Theorem

Basu's Theorem

Bahadur's Theorem

References

I figured out this stuff by myself and for fun. As usual, I found out that others had thought of isomorphic ideas before me:

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