Simulate n ~ N(theta0, 1), plot WBIC / DIC / BPIC against sample size on a log scale, and watch them lock onto the almost-sure limit log(2pi)+1 exactly when the tempering schedule satisfies n*beta_n to infinity. Use the ▸ buttons beside each control (or the Run the experiments launchers) to auto-play; everything runs client-side.

On the large-sample limits of Bayesian model-evaluation statistics

Bayesian information criteria (DIC, BPIC, WBIC) replace a maximised log-likelihood with a posterior expectation of the log-likelihood. This demo reproduces the paper's normal-model experiment (its Figures 1-2 in spirit): simulate n data points from N(θ₀, 1), plot the per-observation criterion against sample size n on a log scale for many replicates, and watch it approach (or fail to approach) the almost-sure limit −2 E[log p(X | θ₀)] = log(2π) + 1. The one thing that decides convergence is the tempering schedule: WBIC locks onto the limit exactly when n β_n → ∞, and stalls above it otherwise.

Bayesian model evaluationWBIC / DIC / BPICPower posteriorsAlmost-sure limits

Read the paper Full text

WBIC convergence vs the tempering schedule β_n

dashed line = a.s. limit log(2π)+1 · dots = replicate criterion values · guide = the n slider · admissible schedules lock on, 1/n & 1/(n log n) stall

n (log) 316 true θ₀ 0.6

WBIC DIC BPIC limit log(2π)+1

The model is the unit-variance normal, p(x | θ) = (2π)^−1/2 exp{−(x−θ)²/2}, with a N(0, 1) prior. The power posterior Π_n^β_n is N(m_n, v_n) with m_n = nβ_nX̄_n/(nβ_n+1) and v_n = 1/(nβ_n+1). The demo plots the paper's closed form (18), WBIC_n = log(2π) + (1/n)∑X_i² − 2X̄_nm_n + m_n² + v_n, per observation, against n. Watanabe's β_n=1/log n, together with 1/log log n, 1, and 1/√n, all satisfy nβ_n→∞ and so WBIC converges almost surely to −2 E[log p(X | θ₀)] = log(2π)+1. Switch to β_n=1/n (then nβ_n=1) or 1/(n log n) (then nβ_n→0) and the condition fails: WBIC stalls above the limit, exactly the paper's point that nβ_n→∞ is necessary. (Honest-draft caveats: this reproduces the normal-model WBIC from the exact closed form (18). Because the data have known unit variance, the a.s. limit log(2π)+1 is θ₀-independent, so the dashed line is fixed and θ₀ only shifts the finite-n shrinkage of the cloud, not the limit. The DIC and BPIC overlays use plausible per-observation forms (posterior deviance plus an effective-parameter penalty v_n) that share the same limit but are not yet matched to the paper's exact penalties; the precise β_n set and a geometric-model panel will be refined against the author's original figure code. Replicate samples are drawn client-side via sufficient statistics.)

Run the experiments

Every animation runs live in your browser. Click a button to run that experiment on the demo (it scrolls up and starts); drag any control to take over. Nothing is downloaded, it is generated on the fly.

Sweep the sample size

Slide n (the star) across the log-scale x-axis and read the expected WBIC and its gap to the limit at each sample size; admissible schedules shrink the gap to zero, the inadmissible ones do not.

Shift the true theta0

Change the data-generating theta0. The a.s. limit log(2pi)+1 stays fixed (it is theta0-independent for unit variance); theta0 only shifts the finite-n shrinkage bias of the replicate cloud.

Tour the tempering schedules

Cycle through beta_n = 1/log n, 1/log log n, 1, 1/sqrt(n), 1/n and 1/(n log n): the first four (n*beta_n to infinity) lock WBIC onto log(2pi)+1, while 1/n and 1/(n log n) stall above it.

Overlay DIC and BPIC

Add the DIC (amber) and BPIC (blue) clouds beside WBIC (teal) to see all three Bayesian criteria collapse onto the same almost-sure limit under an admissible schedule.

Redraw the replicates

Draw fresh replicate samples at every sample size to see the Monte-Carlo scatter of the criterion around its limiting trend.

The idea in three steps

Frequentist information criteria penalise a maximised log-likelihood; their Bayesian cousins penalise a posterior expectation of the log-likelihood instead. The paper asks what these Bayesian criteria actually converge to as data accumulate.

1 · Simulate

Data and a criterion

Draw n points from N(θ₀, 1) and evaluate WBIC in closed form. Repeat over many replicates and sample sizes to trace the criterion as a cloud against n on a log axis.

2 · The limit

An almost-sure target

The paper proves each criterion converges almost surely to −2 E[log p(X | θ₀)], here log(2π)+1: twice the differential entropy of the data-generating law. The dashed line marks it.

3 · The condition

Why the schedule matters

WBIC tempers the posterior by β_n. Convergence holds precisely when nβ_n→∞. Pick 1/n or 1/(n log n) and the cloud stalls above the line, showing the condition cannot be dropped.

For the general almost-sure limits of DIC, BPIC and WBIC, the accompanying posterior and generalised-posterior consistency results, and the technical and numerical examples, see On the large-sample limits of some Bayesian model evaluation statistics (Hien Duy Nguyen, Mayetri Gupta, Jacob Westerhout & TrungTin Nguyen, Econometrics and Statistics 2026).