MMP.7:4 - Solution

Construct the possible recorded outcomes from the observation procedure. Combine the subject and recording laws, remove the unobserved alternatives by the appropriate probability operation, and inspect what the resulting law permits you to infer. Return to the procedure or assumptions when its output does not answer the working question.

MMP.7:4.1 - Choose the target and the recorded outcome separately

State what the answer concerns: a rate in a population, a property before measurement, a future response, or another quantity selected by the work. Specify the population, conditions and time range when they change its meaning. Use C.16 for the characteristic being measured and C.16.MR for the relation from the property to an indication.

Describe one complete outcome of the observing procedure. It may contain a value and an inclusion flag, a duration and a timeout flag, or several related readings. The mathematical outcome space must distinguish every report the procedure can produce that affects the inference.

State what the observation plan fixes. Following a known cohort produces information about excluded cases that a sample drawn only from submitted reports may lack. Stopping after a specified time, after a specified number of records, or after an event can produce different data laws. Recover the actual plan before treating any count as fixed.

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.

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.

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.

MMP.7:4.5 - Test a consequence and revise the construction

Check normalization and a small case that follows the procedure. Enumerate a finite outcome space or generate subject cases and pass them through the recorder. Compare that construction with the probabilities or summaries derived from the observation law. C.29.2 supplies a computational construction when enumeration or integration needs further work.

When the inclusion rule, timeout, shared calibration or receiving question actually changes, revise the affected relation and carry its consequence through the calculation. Compare a plausible alternative condition when the comparison can change the intended use or returned claim; this can expose the observation mechanism beyond one fixed formula. An already sufficient construction under unchanged conditions needs no invented variation.

A simulation agreeing with the formula checks their agreement under the modeled assumptions. An available observation can challenge those assumptions; selected domain assurance determines which empirical comparison is worth performing. C.11.DUA helps choose between further observation, a conditional answer and acting with remaining uncertainty. Preserve a sufficient result without demanding another dataset merely because an influence remains unknown.