# Question 17814: Read One Parameter-Outcome Cell without Confusing Their Roles

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0034-17814 · q-20261014-advance-20261103-r04-parametric-statistical-model-0039



## Original question and givens

Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/17, 1/17, 15/17], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,probability,model_qualified?).


- G001 — Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/17, 1/17, 15/17], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,probability,model_qualified?).





## Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.


- [mko-parametric-statistical-model](https://ocme.evemisslab.com/en/mko/mko-parametric-statistical-model/)





## Step-by-step derivation

S001 · calculationBefore

parameter0 complete outcome mass sum := sum(1/17, 1/17, 15/17)

After

parameter0 complete outcome mass sum = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001





S002 · impliesBefore

original parameter row=0;SAME outcome carrier Fin(3);original aligned rational entries=[[1, 17], [1, 17], [15, 17]]

After

reduced rational candidate entries=[1/17, 1/17, 15/17];sum=1;nonnegative=Yes;row qualified=Yes

Conditions: none

Every parameter needs a complete nonnegative unit-mass law on the same sample space.Invalid candidate rows retain their values and false flags;the name law table does not normalize them.

Step identifiers and dependenciesRule: statmodel_every_parameter_law_same_observation_space · Uses: G001, S001





S003 · calculationBefore

parameter1 complete outcome mass sum := sum(1/2, 1/3, 1/6)

After

parameter1 complete outcome mass sum = 1

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

original parameter row=1;SAME outcome carrier Fin(3);original aligned rational entries=[[1, 2], [1, 3], [1, 6]]

After

reduced rational candidate entries=[1/2, 1/3, 1/6];sum=1;nonnegative=Yes;row qualified=Yes

Conditions: none

Every parameter needs a complete nonnegative unit-mass law on the same sample space.Invalid candidate rows retain their values and false flags;the name law table does not normalize them.

Step identifiers and dependenciesRule: statmodel_every_parameter_law_same_observation_space · Uses: G001, S003





S005 · impliesBefore

original parameter carrier Fin(2);common outcome carrier Fin(3);ALL original row audits=[[0, [1/17, 1/17, 15/17], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]]

After

whole model qualified=Yes;common_sample_space;no parameter prior or posterior supplied

Conditions: none

Conjoin all row qualifications.Parameter indices do not thereby become a probability-distributed random variable.No parameter prior or posterior is supplied;do not normalize across parameters by default.

Step identifiers and dependenciesRule: statmodel_whole_table_qualification_no_parameter_prior · Uses: G001, S004





S006 · impliesBefore

original parameter index=1;original observed-outcome index=1;candidate full row=[1/2, 1/3, 1/6]

After

raw candidate cell=1/3;whole-table qualification remains=Yes

Conditions: none

Select the row by parameter and the column by outcome.A raw cell in an invalid candidate table is not automatically a qualified probability and does not repair the model.

Step identifiers and dependenciesRule: statmodel_original_row_column_cell_not_model_repair · Uses: G001, S005









## Reverse checks


- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute parameter1 complete outcome mass sum with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S001 · premise_and_result_audit — Recompute parameter0 complete outcome mass sum with exact rational arithmetic, not rounded decimals.





## Result

(1, 1, 1/3, Yes)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 7736fd5040d08b813c171479160e75c31bf6503c53cd187a8b1f39d83f873689
- candidate record SHA-256: 1790fabd4c04cfa43d58a7165dfd0d68a54a9c712a08a9500b15bb9c458c723c
- semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336
- question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699
- Lean: NOT_RUN · human review: NOT_PERFORMED

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