A complete model with two parameter values and four sample points

Goal and scope

Validate every parameter law and distinguish likelihood from posterior.

A complete finite table; no posterior or optimality inference.

Givens and symbols

• work

• G1 — Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16.

• θ — fixed unknown parameter

Rules and sources

• mko_definition · cited — Validate the finite parameter set and every nonnegative normalized row of the complete law table. In L(theta;x)=P_theta(x), likelihood varies theta while the observation x is fixed. It is not a posterior; without a prior no Bayes update is performed. One observation does not establish identifiability. · mko-parametric-statistical-model
• work · prerequisite — Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes.

Main derivation

\Theta=\{1/4,3/4\},\quad\Omega=\{00,01,10,11\}

P_\theta:\Omega\to[0,1]\quad(\theta\in\Theta)

P_{1/4}=(9,3,3,1)/16

(9+3+3+1)/16=1

P_{3/4}=(1,3,3,9)/16

(1+3+3+9)/16=1

x=01\text{ fixed}

L(1/4;01)=3/16,\quad L(3/4;01)=3/16

\sum_{\theta\in\Theta}L(\theta;01)

3/16+3/16=3/8\ne1

x=00

L(1/4;00)=9/16>1/16=L(3/4;00)

P_{1/4}(00)=9/16\ne1/16=P_{3/4}(00)

P_{1/4}\ne P_{3/4}

\theta\mapsto P_\theta\text{ with both rows normalized}

\{P_\theta:\theta\in\Theta\}\text{ is a finite parametric model}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, mko_definition — No prior was supplied, so no Bayes update is performed.
• S07 · premise_audit · G1, S06, work — This comparison uses the complete laws, not one observed 01.
• S06 · premise_audit · G1, S05, work — A larger likelihood is not a computed parameter probability.
• S05 · premise_audit · G1, S04, work — The sum3/8 explicitly refutes treating likelihood itself as a posterior.
• S04 · premise_audit · G1, S03, mko_definition — Do not interchange the roles of x and theta.
• S03 · premise_audit · G1, S02, work — These two rows exhaust the parameter set.
• S02 · premise_audit · G1, S01, work — The weight9 is not itself probability 9.
• S01 · premise_audit · G1, work — Both rows require four entries.

Error controls and negative controls

Rowwise

Check both complete denominator 16 rows separately.

Fixed observation

Fix x before traversing every theta for the likelihood table.

• The posterior at 01 is(3/16,3/16). → reject — Its sum is 3/8, not 1, and no prior is supplied.
• Each theta may silently use a different carrier. → reject — This model requires the same four-sample carrier.
• One01 proves the laws identical. → reject — Their00 probabilities differ:9/16 versus1/16.

Result and limitations

The rows define a valid finite parametric model; both L(theta;01) values are 3/16, not a posterior.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: d00df7b6f9cdd89591f628108246c88f499ea10b7d1142e3b840f769915b49dc
• original source SHA-256: fcaef8fae4f87c9c47807672f655dbe899166728be3799d64ec5e07d2e5c74f2
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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