MMP.7:4.3 - Obtain the law for what was actually recorded

The general operation averages the chance of an observed event over the underlying cases. Let K_theta(B given z) be the chance that the report falls in a set B, given underlying value z. Then:

P_theta(O in B) = integral K_theta(B given z) P_theta(dz).

Here P_theta(dz) means averaging with the probability law of Z: a weighted sum for discrete cases or an integral for continuous ones. A deterministic recorder O=g(Z) has K equal to one when g(z) lies in B and zero otherwise. This constructs its output law even when the joint pair (Z,O) has no ordinary joint density, as with O=Z for a continuously varying Z.

When the masses or densities used in :4.2 are available, the same averaging operation gives the law of a particular report. For discrete unobserved alternatives, sum:

p_theta(o) = sum_z p_theta(z) k_theta(o given z).

For continuous alternatives, integrate the product of the subject density and the recording factor. A report produced exactly when Z lies in a fixed set A has recording factor one inside A and zero outside; its probability reduces to the integral of the density over A. If the procedure chooses which set to report, retain that choice in k_theta(A given z).

For example, let Z be equally likely to be 0 or 1. A truthful recorder reports {0,1} always when Z=0 and with probability 1/2 when Z=1; otherwise it reports {1}. The probability of receiving {0,1} is 1/2 + (1/2)(1/2) = 3/4, although the probability that Z lies in {0,1} is one. The recording factor makes the difference.

For an individually observed continuous value, use a density with respect to the stated measurement convention. A point density and the probability of an interval have different meanings.

When inclusion in the dataset is itself a condition of sampling, retain its normalization. If Z has density or mass p_theta(z), and s_theta(z) is its probability of inclusion, the included-case law is:

p_theta(z given included) = p_theta(z) s_theta(z) / P_theta(included).

The denominator is obtained by summing or integrating the numerator over all admitted z and must be positive. If it depends on theta, dropping it changes the inference. When the counts or identities of excluded cases are also observed, include that information in the joint outcome instead of silently discarding it by conditioning. Section :5.1 shows the change.

Keep shared influences shared during elimination. For observations conditionally independent given an unknown B, integrating one joint product over B generally differs from multiplying separately integrated factors. The latter construction assigns a fresh B to each observation. Use it only when that is the observing arrangement.