MMP.7:4.4 - Connect the law to inference and prediction

When the observation laws have a common probability-mass or density representation, insert the recorded outcome o into p_theta(o). As a function of theta, this gives a likelihood, up to a factor independent of theta. It need not sum or integrate to one over theta. Estimation or a posterior distribution requires the chosen inferential method and its assumptions; the observation law is the input to that work.

Before drawing an inference, check whether the recording rule admits the received report for any parameter value. A continuous reading can be admitted even though its single-point probability is zero; determine admissibility from the modeled observation mechanism and the cases it permits. If no admitted case produces the report, return the conflict and locate which assumptions or recording steps need reconsideration, using C.16.IR:4.4. For example, a fixed signal with one fixed additive offset and one unchanged threshold must produce identical bits on repetition. A mixed sequence contradicts that joint account. It cannot be repaired by fitting a different signal within the same family.

Identify how the requested quantity depends on theta or on a future outcome. Two parameter settings can induce the same law for every possible record while assigning different values to the target. Constructing such a pair shows that this observation model cannot identify that distinction. C.16.IR supplies the corresponding compatible-case reasoning; numerical fitting alone cannot resolve it.

For a future record, specify whether its recording procedure is the same. For the underlying population quantity, return through the subject law rather than interpreting a selected-case rate as the population rate. A proposed intervention requires its changed relations under C.28; changing a predictor value in a fitted association is insufficient when the intervention changes how the data arise.