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A variational problem to calculate probabilities

Hugo Guadalupe Reyna-Castañeda, María de los Ángeles Sandoval Romero

TL;DR

The paper develops a functional-analytic approach to conditional expectation by casting $E(X|\mathcal{G})$ as a linear functional on Lebesgue spaces and proving its existence and uniqueness via Riesz–Fréchet representations. It first treats $L^{2}$ with a Dirichlet-type variational perspective and then extends to general $p\in(1,\infty)$ using Riesz representations in $L^{p}$ and $L^{q}$ spaces along with Clarkson inequalities to establish reflexivity. The conditional expectation is characterized as the unique minimizer of an energy functional, and this variational viewpoint is used to prove the law of total probability. Together, these results provide a unified framework for conditional expectation across $L^{p}$ spaces and connect abstract functional analysis with a classical probabilistic identity. The work broadens the toolkit for probabilistic reasoning under conditioning by embedding it in a variational, operator-theoretic setting.

Abstract

In this paper, we prove the existence and uniqueness of the conditional expectation of an event $A$ given a $σ$-algebra $\mathcal{G}$ as a linear problem in the Lebesgue spaces $L^{p}$ associated with a probability space through the Riesz Representation Theorems. For the $L^{2}$ case, we state the Dirichlet's principle. Then, we extend this principle for specific values of $p$, framing the existence of the conditional expectation as a variational problem. We conclude with a proof of the law of total probability using these tools.

A variational problem to calculate probabilities

TL;DR

The paper develops a functional-analytic approach to conditional expectation by casting as a linear functional on Lebesgue spaces and proving its existence and uniqueness via Riesz–Fréchet representations. It first treats with a Dirichlet-type variational perspective and then extends to general using Riesz representations in and spaces along with Clarkson inequalities to establish reflexivity. The conditional expectation is characterized as the unique minimizer of an energy functional, and this variational viewpoint is used to prove the law of total probability. Together, these results provide a unified framework for conditional expectation across spaces and connect abstract functional analysis with a classical probabilistic identity. The work broadens the toolkit for probabilistic reasoning under conditioning by embedding it in a variational, operator-theoretic setting.

Abstract

In this paper, we prove the existence and uniqueness of the conditional expectation of an event given a -algebra as a linear problem in the Lebesgue spaces associated with a probability space through the Riesz Representation Theorems. For the case, we state the Dirichlet's principle. Then, we extend this principle for specific values of , framing the existence of the conditional expectation as a variational problem. We conclude with a proof of the law of total probability using these tools.

Paper Structure

This paper contains 11 sections, 25 theorems, 77 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and some useful facts
  3. Banach spaces
  4. Probability spaces
  5. Problem statement
  6. Problem in $L^2$
  7. Problem in $L^{p}$
  8. The variational problem
  9. Differentiable functionals
  10. Dirichlet's principle
  11. Law of Total Probability Proof

Key Result

Theorem 1.1

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\mathcal{B}=\{B_1,\ldots,B_N\}$ be a collection on $\mathcal{F}$ such that $\Omega=\bigcup_{j=1}^{N} B_j$, $B_{i} \cap B_{j} = \varnothing$ for all $i \neq j$ and $P(B_j) >0$ for all $j=1,\ldots,N$. Then, for any $A \in \mathcal{F}$, we ha where $P(A)$ is the total probability of event $A$.

Figures (5)

  • Figure 1: Law of total probability
  • Figure 2: Partition of $\Omega$
  • Figure 3: Linear combination of indicator functions
  • Figure 4: Orthogonal decomposition of $H$ in the kernel of $L$
  • Figure 5: $J_{u}(t):=\frac{1}{2}\Vert tu \Vert_{2}^{2} - T(tu)$ for fixed $u \in L^{2}(\Omega,\mathcal{G},P)$ with $u \neq 0$ and $t \in \mathbb{R}$

Theorems & Definitions (53)

  • Theorem 1.1: Law of total probability
  • Definition 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Milman-Pettis
  • Corollary 2.7
  • ...and 43 more