Partition evidence before reversing a conditional

Try

A fictional workshop receives 40% of its parts from source A and 60% from B. Their defect probabilities are 0.05 and 0.10. A defective part is found. Predict whether its source probability should equal the production share, the defect rate or a normalized combination of both.

Predict

Scale the model to 1,000 parts. Expected counts are 400 from A and 600 from B. Predict how many defects each source contributes. These are model expectations, not a claim that every actual batch has exactly those counts. Conditioning on defects discards the nondefective branches from the reference group.

Experiment

The expected defect counts are 20 from A and 60 from B, for 80 total. Thus P(defect)=0.08 and P(A|defect)=20/80=1/4. Draw the two source branches and their defect subbranches, then merge only the defect branches into the posterior denominator.

Source	Prior share	Conditional defect rate	Joint defect share
A	0.4	0.05	0.02
B	0.6	0.10	0.06

The prior 0.4 becomes posterior 0.25 because A has the lower defect rate. This is not a contradiction: the reference group changed from all parts to defective parts.

Understand

If H1,H2,… form a finite or countable disjoint exhaustive partition, total probability gives P(E)=ΣP(E|H_i)P(H_i), omitting or handling zero-prior branches appropriately. Each product is a joint probability. Disjointness prevents double counting; exhaustiveness prevents leaving out a possible source of evidence.

For a positive evidence probability, Bayes’ rule is P(H_j|E)=P(E|H_j)P(H_j)/P(E). The numerator is the same joint probability that appears in total probability. The denominator normalizes across every branch leading to E. This formula follows directly from the two ways of writing P(H_j∩E), not from a separate intuition about reversing arrows.

For complementary hypotheses with positive denominators, dividing their posteriors yields posterior odds equal to prior odds times a likelihood ratio. The likelihood ratio P(E|H)/P(E|Hᶜ) measures how the evidence favors one hypothesis relative to the other; the prior odds determine the starting balance. A large likelihood ratio need not imply a large posterior when prior odds are extremely small.

If P(E)=0, the elementary formula does not define a posterior. If a rate is estimated from data, its uncertainty and relevance to the new setting matter. Conditional probabilities may change across populations or operating conditions, so transporting only a base rate while treating every other rate as universal can be unjustified.

Bayes’ rule is a probability identity within a model. It does not itself choose an action, establish the truth of the inputs or turn an observational conditional into a causal intervention. Costs, alternative explanations and model checking remain separate tasks.

For many sources, sum all branches before dividing. Normalizing only against the favored source would produce a value of one by construction and erase the question being asked.

Apply

Rebuild the worked example with a different production mix while holding both conditional defect rates fixed. Explain which quantities change and which are assumed stable. In the rare-screen practice, compare posteriors numerically rather than quoting sensitivity as the answer to a reversed conditional.

Continue

Next assign numerical values to outcomes and describe their full distribution using a cumulative distribution function. This provides one language for discrete, continuous and mixed laws.