# Castedo Ellerman

Proposed answer to the following question(s):

## Distribution Decomposition

Given a set of random variables $$\{ Z_i \}_{i \in I}$$ with countable index set $$I$$, and probability space $$\Omega$$, for each $$i \in I$$, define functions $${B}_{ i}(s)$$ and $$c_i(s)$$ over strings of alphabet $$\{ \mathtt{0}, \mathtt{1} \}$$ [1].

$\begin{eqnarray*} {B}_{ i}(\epsilon) & := & \Omega \\ \end{eqnarray*}$ For any string $$s$$ with $${ \operatorname{P}({ {B}_{ i}(s) }) > 0}$$ , $\begin{eqnarray*} c_i(s) & := & \operatorname{E}\!\left({ Z_i | {B}_{ i}(s) }\right) \\ {B}_{ i}(s \mathtt{0}) & := & {B}_{ i}(s) \cap \{ Z_i < c_i(s) \} \\ {B}_{ i}(s \mathtt{1}) & := & {B}_{ i}(s) \cap \{ Z_i > c_i(s) \} \\ \end{eqnarray*}$ otherwise $\begin{eqnarray*} {B}_{ i}(s\ell) & := & \emptyset \\ \end{eqnarray*}$

A new probability space $$\Omega'$$ is defined by extending $$\Omega$$ such that for each $$i \in I$$ and string $$s \in \{ \mathtt{0}, \mathtt{1} \}^*$$ with $$\operatorname{P}({ {B}_{ i}(s)}) - \operatorname{P}({ \{Z_i = c_i(s)\} }) > 0$$, there are new disjoint events $$C_{i,0}(s)$$ and $$C_{i,1}(s)$$ such that $\begin{eqnarray*} \{Z_i = c_i(s)\} & = & C_{i,0}(s) \cup C_{i,1}(s) \\ \end{eqnarray*}$ and for each $$\ell \in \{ \mathtt{0}, \mathtt{1} \}$$ and all events $$S$$ in $$\Omega$$ $\begin{eqnarray*} \operatorname{P}({ S \cap C_{i,\ell}(s) }) & = & \operatorname{P}({ S \cap \{Z_i = c_i(s)\} }) \frac{ \operatorname{P}({ {B}_{ i}(s\ell)}) }{ \operatorname{P}({ {B}_{ i}(s)})-\operatorname{P}({ \{Z_i = c_i(s)\} }) } \\ \end{eqnarray*}$ Define $\begin{eqnarray*} B'_i(s) & := & {B}_{ i}(s) \cup C_{i,\ell}(s) \\ v_i(s) & = & \sum_{\ell \in \{ \mathtt{0}, \mathtt{1} \}} \operatorname{P}({ {B}_{ i}(s\ell)|{B}_{ i}(s)}) \left( c_i(s\ell)-c(0) \right)^2 \\ h_i(s) & = & \sum_{\ell \in \{ \mathtt{0}, \mathtt{1} \}} h({B}_{ i}(s\ell)|{B}_{ i}(s)) \\ \theta_i & := & \left\{ \{B'_i(s\mathtt{0}),B'_i(s\mathtt{1})\} \mapsto v_i(s)/h_i(s) : \operatorname{P}({ {B}_{ i}(s)}) > 0 \right\} \\ \end{eqnarray*}$

The scope product [2]" across all $$\theta_i$$ is the distribution decomposition of $$\{ Z_i \}_{i \in I}$$.

TODO: scope product across countable index set

# References

1. Hopcroft JE, Ullman JD (1979) Introduction to automata theory, languages, and computation. Addison-Wesley, Reading, Mass

2. Ellerman EC Open study answer #133. https://castedo.com/osa/133/