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Question 586: One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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P1000-0586 · q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0061

Original question and givens

Use complete theta labels=[a10, b10], observations=[x10, y10] 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. Tie order=[a10, b10] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).

  • G001 — Use complete theta labels=[a10, b10], observations=[x10, y10] 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. Tie order=[a10, b10] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).

Definition and scope

Keep each observation's complete argmax, then pick the first member in one fixed parameter-independent tie order, giving a total estimator.

Step-by-step derivation

S001 · implies

Before

Θ=[a10, b10]; Ω=[x10, y10]

After

[[3/4, 1/4], [1/4, 3/4]]

Conditions: none

Each row is one parameter law and each column an outcome; reorder probabilities together with labels.

Step identifiers and dependencies

Rule: full_parametric_model · Uses: G001

S002 · calculation

Before

P_a10(Ω) := sum(3/4, 1/4)

After

P_a10(Ω) = 1

Conditions: none

Check this law's complete total mass first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

P_b10(Ω) := sum(1/4, 3/4)

After

P_b10(Ω) = 1

Conditions: none

Check this law's complete total mass first.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

tie priority=[a10, b10]

After

one fixed ordering, independent of the observed outcome

Conditions: none

Declare one fixed tie order, then choose its first member within each complete argmax.

Step identifiers and dependencies

Rule: fixed_tie_rule · Uses: G001, S003

S005 · implies

Before

parameter=a10; observed=x10

After

L(a10;x10)=3/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependencies

Rule: fixed_likelihood_lookup · Uses: G001, S004

S006 · implies

Before

parameter=b10; observed=x10

After

L(b10;x10)=1/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependencies

Rule: fixed_likelihood_lookup · Uses: G001, S005

S007 · implies

Before

[[a10, 3/4], [b10, 1/4]]

After

maximum=3/4; all argmax=[a10]

Conditions: none

Retain every maximizer; for an all-zero column the complete argmax is still all parameters.

Step identifiers and dependencies

Rule: complete_argmax · Uses: G001, S006

S008 · implies

Before

argmax(x10)=[a10]; priority=[a10, b10]

After

estimator(x10)=a10

Conditions: none

The selector is single-valued; the original argmax need not be a singleton.

Step identifiers and dependencies

Rule: deterministic_tie_selection · Uses: G001, S007

S009 · implies

Before

parameter=a10; observed=y10

After

L(a10;y10)=1/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependencies

Rule: fixed_likelihood_lookup · Uses: G001, S008

S010 · implies

Before

parameter=b10; observed=y10

After

L(b10;y10)=3/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependencies

Rule: fixed_likelihood_lookup · Uses: G001, S009

S011 · implies

Before

[[a10, 1/4], [b10, 3/4]]

After

maximum=3/4; all argmax=[b10]

Conditions: none

Retain every maximizer; for an all-zero column the complete argmax is still all parameters.

Step identifiers and dependencies

Rule: complete_argmax · Uses: G001, S010

S012 · implies

Before

argmax(y10)=[b10]; priority=[a10, b10]

After

estimator(y10)=b10

Conditions: none

The selector is single-valued; the original argmax need not be a singleton.

Step identifiers and dependencies

Rule: deterministic_tie_selection · Uses: G001, S011

S013 · implies

Before

defined on 2 outcomes

After

total on all 2 listed outcomes

Conditions: none

Every listed outcome has an output, including an outcome null under all laws.

Step identifiers and dependencies

Rule: total_estimator · Uses: G001, S012

Reverse checks

  • S013 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S012 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S011 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S010 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S009 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S008 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S007 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S006 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S005 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S004 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
  • S003 · premise_and_result_audit — Recompute P_b10(Ω) with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute P_a10(Ω) with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.

Result

([a10, b10], [[x10, [a10], a10], [y10, [b10], b10]], Yes, theta_independent_tie_rule_makes_a_total_estimator)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.

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: bfba572293185742d75ce8517b5f5314e30323c19f1f78d5b6fd7f73cc69a015
  • candidate record SHA-256: 8a320b3626f37db34760faa9b385615e47fa0ba398fe941d1d132002de60f707
  • semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
  • question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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