ACH ↔ Bayesian Reasoning Explorer

Example:

Question

ACH Table

Click cells to rate consistency

++0.95

+0.80

·0.50

−0.20

−−0.05

Priors P(H)

Your belief before considering any evidence. Defaults to uniform.

Posterior P(H | E)

How probable each hypothesis is after combining the evidence with your priors.

Prior

Posterior

Uniform (—)

Likelihood matrix P(E | H)

The same matrix you filled in above, translated into numbers. Each ACH consistency rating is a likelihood — the probability of seeing that evidence if the hypothesis were true. Darker = more likely.

0.00

1.00 P(E | H)

Posterior contributions (log-likelihood)

Each cell shows how much that evidence pushes belief toward or away from each hypothesis, relative to the row average. Add down each column (plus the log-prior) to get the posterior ranking.

−1.00

+1.00 Δ log-likelihood vs. row mean

The Bayesian correspondence

An ACH table is a Bayesian inference in disguise. Each consistency rating in a cell is an implicit likelihood P(Eᵢ | Hⱼ) — how probable that evidence would be if the hypothesis were true. Combining the rows assumes the pieces of evidence are conditionally independent given the hypothesis (the naive Bayes assumption), giving the unnormalized joint:

P(Hⱼ | E₁, …, Eₙ) ∝ P(Hⱼ) · P(E₁ | Hⱼ) · P(E₂ | Hⱼ) · … · P(Eₙ | Hⱼ)

Normalizing across hypotheses gives the posteriors shown above. The evidence rows with high diagnosticity — where likelihoods differ most across hypotheses — do the most work in moving the posterior away from the prior.

ACH ↔ Bayesian Reasoning

ACH Table

Priors P(H)

Posterior P(H | E)

Likelihood matrix P(E | H)

Posterior contributions (log-likelihood)

The Bayesian correspondence

Likelihood matrix P(Eᵢ | Hⱼ)

Unnormalized joint P(Hⱼ) · ∏ᵢ P(Eᵢ | Hⱼ)

Posterior P(Hⱼ | E)