MMP.8:4.2 - Recover the order and availability of information

Describe what is observed before each decision and what arrives afterward. Express the available observation as h(w) when it is a deterministic description of the circumstance. It may reveal only part of w. For a random or noisy observation, specify its probability law for each admitted w. An unknown fixed w can index a family P_w of observation laws without having a probability distribution itself. A joint law is needed when the question also treats w as random and averages over it. MMP.7 constructs the recording law.

A policy pi selects an action from the information available: a=pi(h(w)). Two circumstances with the same h(w) must therefore receive the same action. This uses MATH.2’s condition that an answer remains constant on cases identified by a description. Here the observation determines those groups. The information restriction is often called nonanticipativity: the policy does not use distinctions the decision-maker has yet to observe.

For repeated interaction, use the observation history available at each decision. A policy may remember earlier observations or actions. Omitting that memory is a substantive model choice. Global termination is unnecessary when the question concerns a continuing response; specify the response or progress condition that matters under the admitted inputs.

If an earlier action changes what can be observed, make h depend on that action and describe the cost and timing of observation. If action also changes the circumstance distribution or evolution, include that relation. C.28 supplies the causal-use question and the policy’s permitted pre-action information; C.28.MR constructs the mechanism replacement. A changed distribution cannot be inferred from a favorable selection of historical cases alone.