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P250-0167 · q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0042
Use complete theta labels=[a16, b16], observations=[x16, y16] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x16. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).
G001 — Use complete theta labels=[a16, b16], observations=[x16, y16] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Observed label=x16. Give (observed, likelihood_rows, maximum_likelihood, argmax_parameters, unique_argmax, certificate).Use complete finite parameter and observation carriers. Fix the observed column, retain the full argmax, and introduce no prior.
Before
Θ=[a16, b16]; Ω=[x16, y16]
After
observed=x16, column=0
Conditions: none
Fix the observed outcome and compare its one probability column across parameter values.
Rule: fixed_observation · Uses: G001
Before
P_a16(Ω) := sum(3/4, 1/4)
After
P_a16(Ω) = 1
Conditions: none
First check that this parameter's row remains a normalized probability law.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
row for a16=[3/4, 1/4]
After
L(a16;x16)=P_a16(x16)=3/4
Conditions: none
Take the probability at the observed label in this row, without introducing a prior.
Rule: likelihood_lookup · Uses: G001, S002
Before
P_b16(Ω) := sum(1/4, 3/4)
After
P_b16(Ω) = 1
Conditions: none
First check that this parameter's row remains a normalized probability law.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
row for b16=[1/4, 3/4]
After
L(b16;x16)=P_b16(x16)=1/4
Conditions: none
Take the probability at the observed label in this row, without introducing a prior.
Rule: likelihood_lookup · Uses: G001, S004
Before
L(a16;x16)=3/4, maximum=3/4
After
a16 ∈ argmax: Yes
Conditions: none
Include every parameter that attains the maximum, including ties after the first maximizer.
Rule: argmax_membership · Uses: G001, S005
Before
L(b16;x16)=1/4, maximum=3/4
After
b16 ∈ argmax: No
Conditions: none
Include every parameter that attains the maximum, including ties after the first maximizer.
Rule: argmax_membership · Uses: G001, S006
Before
argmax=[a16]
After
unique maximizing parameter: Yes
Conditions: none
Only a complete argmax with exactly one element has a unique maximizing parameter.
Rule: cardinality_one · Uses: G001, S007
Before
finite normalized laws
After
MLE is not a posterior or an automatic quality certificate
Conditions: none
If all likelihoods at an observation are zero, every parameter still maximizes at 0; this computation does not imply unbiasedness or consistency.
Rule: scope_boundary · Uses: G001, S008
S009 · premise_and_result_audit — Do not confuse a null observation column with an invalid all-zero probability-law row.S008 · premise_and_result_audit — A fixed tie-breaker can select one output without making a multi-element argmax a singleton.S007 · premise_and_result_audit — The maximum is a number; argmax is a set of parameters.S006 · premise_and_result_audit — The maximum is a number; argmax is a set of parameters.S005 · premise_and_result_audit — The value remains bound to the same parameter and observation.S004 · premise_and_result_audit — Recompute P_b16(Ω) with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — The value remains bound to the same parameter and observation.S002 · premise_and_result_audit — Recompute P_a16(Ω) with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Do not confuse other observations in a row with other parameter values.(x16, [[a16, 3/4], [b16, 1/4]], 3/4, [a16], Yes, MLE_argmax_need_not_be_unique)
The maximum likelihood is 3/4; the complete argmax is [a16].
Review date: 2026-10-24
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
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