Bayes reversal in a four-atom test model

Goal and scope

Compute priors, likelihoods and posteriors from a complete joint law.

A synthetic finite model, with no medical or causal inference.

Givens and symbols

• work

• G1 — Omega={D+,D-,N+,N-} has masses 1/20,1/20,9/100,81/100; D={D+,D-}, N is its complement, E={D+,N+}.

• D,N — disjoint hypotheses
• E — evidence event

Rules and sources

• mko_definition · cited — Use exact rational masses in one complete finite atomic law. Both the two-event and partition forms require positive conditioning denominators. Partition hypotheses must be disjoint and exhaustive, and the evidence mass obtained by total probability must be positive. Bayes conditioning is not causation. · mko-bayes-theorem
• work · prerequisite — For positive denominators P(A|B)=P(A∩B)/P(B); finite additivity over a complete disjoint partition gives total probability.

Main derivation

1/20+1/20+9/100+81/100

1

D\cap N=\varnothing,\quad D\cup N=\Omega

P(D)=1/10,\quad P(N)=9/10

P(E\cap D)=1/20

P(E\mid D)=(1/20)/(1/10)=1/2

P(E\cap N)=9/100

P(E\mid N)=(9/100)/(9/10)=1/10

P(E)=P(E\mid D)P(D)+P(E\mid N)P(N)

P(E)=1/20+9/100=7/50>0

P(D\mid E)=P(D\cap E)/P(E)

P(D\mid E)=(1/20)/(7/50)=5/14

P(N\mid E)=(9/100)/(7/50)

P(N\mid E)=9/14,\quad5/14+9/14=1

P(E\mid D)P(D)/P(E)

\frac{(1/2)(1/10)}{7/50}=5/14

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — The result describes a conditional law, not a causal effect.
• S07 · premise_audit · G1, S06, work — Both posterior entries condition on the same evidence event.
• S06 · premise_audit · G1, S05, work — The posterior 5/14 differs from likelihood 1/2; direction was not silently equated.
• S05 · premise_audit · G1, S04, work — Both likelihoods retain their priors, and positivity is established.
• S04 · premise_audit · G1, S03, work — N has prior 9/10, not D's1/10.
• S03 · premise_audit · G1, S02, work — The denominator 1/10 is strictly positive.
• S02 · premise_audit · G1, S01, work — The atom sets certify a partition, not independence.
• S01 · premise_audit · G1, work — Listing only positive-test atoms would omit part of the law.

Error controls and negative controls

Complete partition

Audit disjointness and exhaustiveness before total probability.

Direction

Use the common joint numerator and distinct positive denominators to distinguish directions.

• P(D|E)=P(E|D)=1/2. → reject — The priors and evidence give 5/14.
• Posterior is proportional to likelihood without a prior. → reject — The unequal priors 1/10 and 9/10 cannot be omitted.
• The Bayes quotient also applies when P(E)=0. → reject — The conditioning denominator must be strictly positive.

Result and limitations

P(D|E)=5/14 and P(N|E)=9/14; neither is a causal conclusion.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 7e8ec8d8f62c0d717ec5d6930e0b5c893570e5fc01e96107d2673d7011ee5d2a
• original source SHA-256: 74781eca7056072aa2c5331410f123e33209127243c5a4c6fb42825494c246dd
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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