MMP.7:4.2 - Construct the joint law from the modeled dependencies

Introduce variables for the quantities used by that procedure. Explain their domains and meanings before assigning distributions. Let Z denote an underlying event or value and O its recorded outcome. Parameters theta describe quantities held fixed in the proposed probability model. When these laws are represented by probability masses or by densities under an appropriate reference measure, write the subject law as p_theta(z) and the conditional recording law as k_theta(o given z). Their joint expression is:

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

Each factor needs an interpretation. The first describes variation in the subject under the stated conditions; the second describes how the procedure records it. A deterministic recorder assigns probability one to its specified output and zero to the other outputs. This accommodates rounding and threshold reports as well as random response or selection.

Use a sequence of conditional laws when more stages matter. Multiplication follows the chain rule. Omitting a variable from a conditional law asserts that, given the retained variables, it does not change that law. Make that assumption from the modeled relation; separate rows in a file provide no independence argument.

Keep an unknown fixed parameter as unknown. Give it a probability distribution only when that additional modeling choice is justified for the intended inference. A shared but unknown calibration offset can remain a parameter in a joint likelihood. A distribution over possible offsets supports a different, explicitly extended model.