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R0034-17833 · q-20261014-advance-20261103-r04-parametric-statistical-model-0058
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=[[1/11, 1/11, 9/11], [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. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.
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=[[1/11, 1/11, 9/11], [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. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.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.
Before
parameter0 complete outcome mass sum := sum(1/11, 1/11, 9/11)
After
parameter0 complete outcome mass sum = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Rule: exact_rational_arithmetic · Uses: G001
Before
original parameter row=0;SAME outcome carrier Fin(3);original aligned rational entries=[[1, 11], [1, 11], [9, 11]]
After
reduced rational candidate entries=[1/11, 1/11, 9/11];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.
Rule: statmodel_every_parameter_law_same_observation_space · Uses: G001, S001
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
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.
Rule: statmodel_every_parameter_law_same_observation_space · Uses: G001, S003
Before
original parameter carrier Fin(2);common outcome carrier Fin(3);ALL original row audits=[[0, [1/11, 1/11, 9/11], 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.
Rule: statmodel_whole_table_qualification_no_parameter_prior · Uses: G001, S004
Before
FIXED observed outcome=1;varying parameter=0;original full row=[1/11, 1/11, 9/11]
After
selected SAME outcome column value=1/11
Conditions: none
Fix the observation and vary the parameter while reading the same column.This is not a different-outcome row function or a conditioned posterior.
Rule: statmodel_likelihood_fixed_observation_varies_parameter · Uses: G001, S005
Before
FIXED observed outcome=1;varying parameter=1;original full row=[1/2, 1/3, 1/6]
After
selected SAME outcome column value=1/3
Conditions: none
Fix the observation and vary the parameter while reading the same column.This is not a different-outcome row function or a conditioned posterior.
Rule: statmodel_likelihood_fixed_observation_varies_parameter · Uses: G001, S006
Before
likelihood column sum not a parameter probability law := sum(1/11, 1/3)
After
likelihood column sum not a parameter probability law = 14/33
Conditions: none
Compute exactly from original roles and established intermediates,retaining denominators,signs,indices and strict conditions.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
fixed observed outcome=1;ordered parameter likelihood rows=[[0, 1/11], [1, 1/3]];column sum=14/33
After
likelihood_as_function_of_parameter_not_posterior;column need not sum1,and even accidental sum1 creates no prior/posterior
Conditions: none
A likelihood is a function of the parameter and need not sum to1 over parameters.Even accidental normalization does not make it a posterior.Do not invent a prior or divide by the column sum to rewrite the question.
Rule: statmodel_likelihood_not_posterior_or_implicit_normalization · Uses: G001, S008
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute likelihood column sum not a parameter probability law with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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.(1, [[0, 1/11], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: fea166a6d7eee8b1b068db13c659986e36d1e356ce51cdcd3600d489a1de6c11candidate record SHA-256: dc7a836c9d91fdfcad353c1ebad37fce2d9bfa56788621d27e05e602a7412f44semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699Lean: NOT_RUN · human review: NOT_PERFORMED