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”