bayesFactorCost_at_null
plain-language theorem explainer
When likelihood equals prior the Bayes factor cost is zero. Bayesian statisticians cite this to fix the zero-information point in Recognition Science models of belief updating. The proof is a one-line wrapper that unfolds the definition, rewrites the ratio to unity, and applies the J-cost unit lemma.
Claim. For any nonzero real $p$, if the likelihood equals the prior then the J-cost of their ratio is zero: $Jcost(p/p)=0$.
background
Jcost is defined in the Cost module as $Jcost(x)=(x-1)^2/(2x)$, which equals zero precisely at $x=1$. The sibling definition bayesFactorCost(likelihood, prior) := Jcost(likelihood/prior) therefore measures information gain on the likelihood ratio. The module sets this inside Bayesian updating where posterior is proportional to likelihood times prior and the KL divergence from prior to posterior quantifies the update step. The upstream lemma Jcost_unit0 states Jcost(1)=0 and is invoked directly here.
proof idea
One-line wrapper that unfolds bayesFactorCost, rewrites the ratio to 1 via div_self on the nonzero hypothesis, and applies the exact lemma Jcost_unit0.
why it matters
This anchors the zero point of the cost function inside the BayesianUpdateCert record, which collects the null case, nonnegativity, and the moderate-evidence thresholds. It directly supports the module's claim that a J-cost of order J(phi) marks the minimum detectable update. The result closes the null case in the structural theorem that links Recognition Science J-cost to Kass-Raftery Bayes-factor bands.
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