Question 17843: A Likelihood Varies the Parameter for One Fixed Observation

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=[[1/21, 1/21, 19/21], [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=2. 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=[[1/21, 1/21, 19/21], [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=2. 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

Step-by-step derivation

parameter0 complete outcome mass sum := sum(1/21, 1/21, 19/21)

parameter0 complete outcome mass sum = 1

original parameter row=0;SAME outcome carrier Fin(3);original aligned rational entries=[[1, 21], [1, 21], [19, 21]]

reduced rational candidate entries=[1/21, 1/21, 19/21];sum=1;nonnegative=Yes;row qualified=Yes

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

parameter1 complete outcome mass sum = 1

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

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

original parameter carrier Fin(2);common outcome carrier Fin(3);ALL original row audits=[[0, [1/21, 1/21, 19/21], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]]

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

FIXED observed outcome=2;varying parameter=0;original full row=[1/21, 1/21, 19/21]

selected SAME outcome column value=19/21

FIXED observed outcome=2;varying parameter=1;original full row=[1/2, 1/3, 1/6]

selected SAME outcome column value=1/6

likelihood column sum not a parameter probability law := sum(19/21, 1/6)

likelihood column sum not a parameter probability law = 15/14

fixed observed outcome=2;ordered parameter likelihood rows=[[0, 19/21], [1, 1/6]];column sum=15/14

likelihood_as_function_of_parameter_not_posterior;column need not sum1,and even accidental sum1 creates no prior/posterior

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

(2, [[0, 19/21], [1, 1/6]], 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.