Bayes update rule studied as an operator

Question

Bayes' rule can be understood as a nonlinear map from the space of probability measures to itself. Are there any reference/books which study it from this perspective? E.g., can we say something about whether it is contractive or not, behaviour under iteration etc. Bayes' rule also induces a stochastic process that is both a martingale and a Markov chain. Any references along these lines would also be interesting.

Answer 1

There is a very large literature that touches on the core properties of the Bayesian updating mapping. Some of this literature is statistical and some of it goes further into the domain of philosophy. The former tends to look at the mathematical properties of Bayesian updating of various kinds of models whereas the latter tends to look at underlying issues relating to whether or not inferences under evidence should (or should not) obey standard properties of Bayes' rule (e.g., commutativity, associativity, etc.). There are two well-known strands of statistical/philosophical literature that look at the properties of iterative application of Bayes' rule (the literature given really just scratches the surface):

• (1) Literature that examines the commutativity/associativity of Bayesian updating for sequences of data updates, "order effects" for how to represent updating when the order of observation of data is considered to affect its interpretation, and philosophical issues relating to desiderata for evidentiary updating (see e.g., Itzhak and David 1993, Weisberg 2009, Wilson 2010, Gardiner 2014, Cassell 2020, Cassell 2021).

• (2) Literature on the consistency/inconsistency properties of Bayesian updating --- e.g., looking at the "frequentist consistency" of Bayesian analysis under correct or incorrect model specification in a range of model classes (see e.g., Berk 1970, Diaconis and Freedman 1986, Barron, Schervish and Wasserman 1999, Ghosal, Ghosh and van der Vaart 2000, Shen and Wasserman 2001, Walker 2009, Shalizi 2009).

As to the idea of examining Bayes' rule "from the perspective" of it being a mapping or Markov chain on underlying probability distributions, this is really implicit in all investigations of the general properties of Bayes' rule. The Markovian property for a sequence of data updates comes from the underlying commutativity and associativity of the operation, so it ties back into statistical/philosophical literature that looks at these properties. The fact that the rule is a "self-map" from the set of probability distributions onto itself is an implicit aspect of most of this work, but sometimes this fact is stated explicitly.

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Bayesian updating as a mapping: Although Bayesian updating is not strictly a linear mapping, it can be represented in a way that is closely linked to an underlying linear mapping. To do this, suppose we consider the set $\Pi$ composed of all probability measures on some sample space of interest and let $\Sigma$ denote the larger set of all sigma-finite measures on that same sample space. Then each probability measure $\pi \in \Pi$ stands in a one-to-one relationship with an equivalence class of sigma-finite measures $\Sigma_\pi \equiv \{ \sigma \in \Sigma | \sigma \propto \pi \}$, which is composed of all of the sigma-finite measures that are proportionate to $\pi$.

Suppose we let $N: \Sigma \rightarrow \Pi$ denote the norming function which maps from any sigma-finite measure to the corresponding probability measure (by norming the measure to a total norm of one). For a given likelihood function $L_\mathbf{x}$, Bayes' rule can be written as the operator $B: \Pi \rightarrow \Pi$ given by:

$$B(\pi) = N(B_*(\pi)) \quad \quad \quad \quad \quad B_*(\pi) = L_\mathbf{x} \cdot \pi.$$

The function $B_*$ is a linear mapping so the only nonlinear part is the norming operation at the end. Alternatively, we can think of the problem in terms of underlying sigma-finite measures and let any $\sigma \in \Sigma$ be a "representative" of the equivalence class giving a single probability measure. In this case, we can write Bayes' rule as the linear operator $B_*: \Sigma \rightarrow \Sigma$ given by:

$$B_*(\sigma) = L_\mathbf{x} \cdot \sigma,$$

and we take the inputs and outputs of Bayes' rule to be "representatives" of probability measures (with the norming part removed) with the equivalence classes on $\Sigma$ used to see if we are dealing with effectively the same input/output. As you can see, although this means that Bayesian updating is not a linear mapping, it is "closely related" to a linear mapping.