Bayesian testing compares posterior probabilities of hypotheses or models. Asymptotically, the answer depends on testing separation, prior mass, and the dimensional penalty in marginal likelihoods.
Posterior odds
Testing is model comparison when the hypotheses are assigned prior probability.
Definition. For hypotheses \(H_0,H_1\), posterior odds are \[ \frac{\Pi(H_1\mid X)}{\Pi(H_0\mid X)} =\frac{\Pi(H_1)}{\Pi(H_0)} \frac{m_1(X)}{m_0(X)}, \] where \(m_j\) is the marginal likelihood under \(H_j\).
Proof. Apply Bayes formula to the binary model index. The likelihood of each hypothesis after integrating nuisance parameters is its marginal likelihood.
Testing consistency
A Bayesian test is consistent when posterior mass moves to the true hypothesis.
Proposition. If under \(H_0\), \(m_1(X)/m_0(X)\to0\) in probability, then \(\Pi(H_1\mid X)\to0\) for fixed positive prior odds.
Proof. Posterior odds equal prior odds times the Bayes factor. Fixed prior odds cannot prevent convergence to zero of the Bayes factor.
Occam penalties
Bayesian model selection naturally penalizes dimension through integration.
Proposition. In regular nested models, Laplace approximation gives a marginal likelihood penalty of order \(n^{-d/2}\) for \(d\) free local dimensions.
Proof. The local posterior volume around the maximum likelihood estimator has radius \(n^{-1/2}\) in each of \(d\) directions. The Gaussian integral therefore contributes \(n^{-d/2}\).
Point nulls and continuous priors
Testing a point null requires assigning it positive prior mass as a model, not merely using a continuous prior over the full parameter.
Remark. If a continuous prior assigns zero mass to \(\theta=\theta_0\), then the posterior probability of the point null remains zero. Bayesian point-null testing therefore uses a mixture prior with an atom at the null and a continuous alternative component.
Posterior odds and exponential separation
Theorem (simple hypotheses). Let \(H_0:P=P_0\) and \(H_1:P=P_1\), both with positive prior mass. Under \(P_0\), \[ \frac1n\log \frac{\Pi(H_1\mid X_{1:n})} {\Pi(H_0\mid X_{1:n})} \longrightarrow -K(P_0,P_1) \quad\text{almost surely}. \] If \(P_0\ne P_1\), posterior probability of \(H_1\) therefore decays exponentially.
Proof. Posterior log odds equal constant prior log odds plus \[ \sum_{i=1}^n \log\frac{p_1(X_i)}{p_0(X_i)}. \] The strong law sends the average log likelihood ratio to \[ E_0\log(p_1/p_0) =-K(P_0,P_1). \]
Composite testing with tests and prior mass
Theorem. Suppose the truth lies in model \(M_0\), the prior gives \(M_0\) positive probability and KL mass near truth, and there are exponentially consistent tests separating \(P_0\) from all distributions in \(M_1\). Then \[ \Pi(M_1\mid X^{(n)}) \xrightarrow{P_0}0. \]
Proof. Bound the integrated likelihood over \(M_1\) by splitting according to the test. Its type-II bound controls the un-rejected part after change of measure; type-I error controls the rejected part under truth. A KL neighborhood inside \(M_0\) lower-bounds the posterior denominator. Their ratio vanishes.
Nested regular models and dimension penalty
Theorem. Let \(M_0\subset M_1\) be regular nested models of dimensions \(d_0<d_1\), with fixed proper priors positive near truth.
- If \(M_0\) is true, \[ \log BF_{10} =-\frac{d_1-d_0}{2}\log n+O_P(1), \] so \(BF_{10}\to0\).
- If truth belongs to \(M_1\) at positive KL distance from \(M_0\), \[ \frac1n\log BF_{10}\to \inf_{\theta_0\in M_0}K(P^\star,P_{\theta_0})>0, \] up to sign convention, so \(M_1\) wins exponentially.
Proof. Under the null, maximized log-likelihood improvement is \(O_P(1)\), while Laplace approximation contributes \(-\{d_1-d_0\}\log n/2\). Under a fixed alternative, the law of large numbers makes the log-likelihood gap order \(n\) with coefficient equal to the KL projection gap; this dominates logarithmic complexity.
Spike-and-slab point-null posterior
Example. Let \[ \theta\sim w_0\delta_0+(1-w_0)N(0,\tau^2), \qquad \bar X\mid\theta\sim N(\theta,1/n). \] Then \[ P(\theta=0\mid\bar X) =\frac{w_0\varphi(\bar X;0,1/n)} {w_0\varphi(\bar X;0,1/n) +(1-w_0)\varphi(\bar X;0,\tau^2+1/n)}. \] Conditional on the slab, the posterior is normal with variance \[ v_n=(n+\tau^{-2})^{-1} \] and mean \(v_nn\bar X\). Model uncertainty and within-model parameter uncertainty are distinct posterior components.
Multiplicity correction through model priors
Proposition. In variable selection with \(p\) predictors, assign model size \(s\) prior \(\pi(s)\) and choose a subset uniformly conditional on \(s\). Then each model \(S\) of size \(s\) has prior \[ \Pi(S)=\frac{\pi(s)}{\binom ps}. \] For fixed \(s\ll p\), \[ -\log\Pi(S) \asymp s\log(p/s)-\log\pi(s), \] which supplies an automatic multiplicity penalty.
Proof. The first identity is conditional probability. Stirling's approximation gives \[ \log\binom ps =s\log(p/s)+O(s) \] for sparse \(s\).
Selection consistency and beta-min
Theorem (sparse selection template). In a sparse regular regression model, posterior model-selection consistency \[ \Pi(S=S_0\mid X^{(n)})\to1 \] requires, besides compatible design and suitable model-size priors, a signal condition of the form \[ \min_{j\in S_0}|\beta_{0j}| \gtrsim \sqrt{\frac{\log p}{n}}. \] Without separation above the testing noise level, uniform exact support recovery is impossible.
Reason. Distinguishing a coefficient zero from \(\delta_n\) is a two-point testing problem. In a normalized regular design, KL divergence is order \(n\delta_n^2\). Searching among \(p\) coordinates costs order \(\log p\). If \(n\delta_n^2\ll\log p\), Fano or multiple-testing lower bounds prevent uniformly reliable localization. Above that scale, Bayes factors can overcome model multiplicity under compatible priors.
Prediction and selection diverge
A posterior can contract at an optimal predictive rate while spreading mass over many nearly equivalent models. Exact support recovery is stronger than estimating \(X\beta\), and correlated predictors can make the “true model” unstable or scientifically unhelpful. Bayesian model averaging targets posterior prediction across models; median-probability or decision-theoretic selection targets a chosen action. These goals need not return the same subset.
Nonlocal priors
Definition. A nonlocal slab prior vanishes when an alternative parameter approaches the null, for example \[ \pi(\theta\mid H_1) \propto\theta^{2r}N(\theta;0,\tau^2), \qquad r\ge1. \] It increases the asymptotic penalty on alternatives whose parameters lie near the null and can accelerate deletion of spurious variables. It also encodes a substantive gap between null and meaningful effects and can reduce power for genuinely small signals.
Testing claims must always report the model prior, slab scale, and whether asymptotics concern fixed alternatives, local alternatives, or growing model spaces. “Bayesian consistency” without that regime is incomplete.