Algebraic Probability

The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities.

In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space ${(\Omega,\mathcal F,{\mathbb P})}$. This consists of a state space ${\Omega}$, an event space ${\mathcal F}$, which is a sigma-algebra of subsets of ${\Omega}$, and a probability measure ${{\mathbb P}}$. The measure ${{\mathbb P}}$ is defined as a map ${{\mathbb P}\colon\mathcal F\rightarrow{\mathbb R}^+}$ satisfying countable additivity and normalised as ${{\mathbb P}(\Omega)=1}$.

A measure space allows us to define integrals of real-valued measurable functions or, in the language of probability, expectations of random variables. We construct the set ${L^\infty(\Omega,\mathcal F)}$ of all bounded measurable functions ${X\colon\Omega\rightarrow{\mathbb R}}$. This is a real vector space and, as it is closed under multiplication, is an algebra. Expectation, by definition, is the unique linear map ${L^\infty\rightarrow{\mathbb R}}$, ${X\mapsto{\mathbb E}[X]}$ satisfying ${{\mathbb E}[1_A]={\mathbb P}(A)}$ for ${A\in\mathcal F}$ and monotone convergence: if ${X_n\in L^\infty}$ is a nonnegative sequence increasing to a bounded limit ${X}$, then ${{\mathbb E}[X_n]}$ tends to ${{\mathbb E}[X]}$.

In the opposite direction, any nonnegative linear map ${p\colon L^\infty(\Omega,\mathcal F)\rightarrow{\mathbb R}}$ satisfying monotone convergence and ${p(1)=1}$ defines a probability measure by ${{\mathbb P}(A)=p(1_A)}$. This is the unique measure with respect to which expectation agrees with the linear map, ${{\mathbb E}=p}$. So, probability measures are in one-to-one correspondence with such linear maps, and they can be viewed as one and the same thing. The Kolmogorov definition of a probability space can be thought of as representing the expectation on the subset of ${L^\infty}$ consisting of indicator functions ${1_A}$. In practice, it is often more convenient to start with a different subset of ${L^\infty}$. For example, probability measures on ${{\mathbb R}^+}$ can be defined via their Laplace transform, ${\mathcal L_{{\mathbb P}}(a)=\int e^{-ax}d{\mathbb P}(x)}$, which represents the expectation on exponential functions ${x\mapsto e^{-ax}}$. Generalising to complex-valued random variables, probability measures on ${{\mathbb R}}$ are often represented by their characteristic function ${\varphi(a)=\int e^{iax}d{\mathbb P}(x)}$, which is just the expectation of the complex exponentials ${x\mapsto e^{iax}}$. In fact, by the monotone class theorem, we can uniquely represent probability measures on ${(\Omega,\mathcal F)}$ by the expectations on any subset ${\mathcal K\subseteq L^\infty}$ which is closed under taking products and generates the sigma-algebra ${\mathcal F}$. Continue reading “Algebraic Probability”