# Question 17838: A Likelihood Varies the Parameter for One Fixed Observation [Return to catalogue page 90](https://ocme.evemisslab.com/en/questions/page/90/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations/#q-20261014-advance-20261103-r04-parametric-statistical-model-0063) 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. R0034-17838 · q-20261014-advance-20261103-r04-parametric-statistical-model-0063 ## 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=[[1/16, 1/16, 14/16], [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=0. 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=[[1/16, 1/16, 14/16], [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=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order. ## 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(1/16, 1/16, 7/8) 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, 16], [1, 16], [14, 16]] After reduced rational candidate entries=[1/16, 1/16, 7/8];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, [1/16, 1/16, 7/8], 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 FIXED observed outcome=0;varying parameter=0;original full row=[1/16, 1/16, 7/8] After selected SAME outcome column value=1/16 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. Step identifiers and dependenciesRule: statmodel_likelihood_fixed_observation_varies_parameter · Uses: G001, S005 S007 · impliesBefore FIXED observed outcome=0;varying parameter=1;original full row=[1/2, 1/3, 1/6] After selected SAME outcome column value=1/2 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. Step identifiers and dependenciesRule: statmodel_likelihood_fixed_observation_varies_parameter · Uses: G001, S006 S008 · calculationBefore likelihood column sum not a parameter probability law := sum(1/16, 1/2) After likelihood column sum not a parameter probability law = 9/16 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, S007 S009 · impliesBefore fixed observed outcome=0;ordered parameter likelihood rows=[[0, 1/16], [1, 1/2]];column sum=9/16 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. Step identifiers and dependenciesRule: statmodel_likelihood_not_posterior_or_implicit_normalization · Uses: G001, S008 ## Reverse checks - 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. ## Result (0, [[0, 1/16], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior) 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: 760963d44169347f99bc5d56cc4a272c173dd87364679c996288235aee1f9319 - candidate record SHA-256: f62e1a6812ac24563d07051803dd06035449c04b5d049e3a057faf435e913a17 - semantic review receipt SHA-256: f73d099ebe3ff1c69a5d96f08c75b11b9e5b05120af40cf3ef388fce3b5c5336 - question-first witness SHA-256: 22675ffb6209b4fea076032b29b96e3dc19aad4244c54493f5f65f5e5d875699 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261103-r04-parametric-statistical-model-0063.json)