Calibrated Uncertainty Quantification for Predictive Bayesian Inference

The talk in one slide

Question. Predictive Bayes replaces likelihood and prior with an engine for future data. Are its credible sets calibrated?

Finding. The engine, not the sampling variability, sets the width: under misspecification a 95% interval covers 70–86% in sparse regression, and under half the time for an upper quartile of skewed data.

Fix. Start the paths from bootstrap resamples: calibrated at the same budget, 28 s where BayesBag needs 23 min.

Plan. Latent-structure engines, then predictive Bayes with the PrO posterior.

Roadmap: 1 Predictive Bayes and the question · 2 What the MGP measures · 3 The fix · 4 Evidence and its limits · 5 Plan

1 · Predictive Bayes and the question

Uncertainty is “due to missing observations. If a complete population is observed, then the parameter of interest can be assumed to be known precisely” (Fong, Holmes, and Walker 2023)

Predictive Bayes replaces the likelihood–prior pair with a predictive engine

Bayes needs a model f_\theta and a prior \pi(\theta): \pi_n(\theta)\propto\pi(\theta)f_\theta(x_{1:n}), usually by MCMC.

Predictive Bayes (Berti et al. 2021; Fortini and Petrone 2025; Fong, Holmes, and Walker 2023) needs one rule for the next observation, the predictive engine \mathsf F(X_{n+1}\in\cdot\mid x_{1:n}).

observed data
x_{1:n}

predictive engine
\mathsf F(\cdot\mid x_{1:n})

simulated future
z_{n+1},\dots,z_N

functional
\vartheta=T(\mathbb P_{n,N})

martingale
posterior \Pi_N

In use for quantile estimation (Fong and Yiu 2026b), neural processes (Lee et al. 2023) and tabular foundation models (Ng et al. 2026); little is known about whether the uncertainty it reports is calibrated.

Parameter uncertainty is whatever the engine believes about the unobserved future. That is the strength, and the risk.

Two ingredients become one. The engine may be a parametric update, a copula update, a neural network. No MCMC, so the approach is spreading fast: quantile regression (Fong and Yiu 2026b), neural processes (Lee et al. 2023), tabular foundation models (Ng et al. 2026). Nothing in the construction refers to a sampling distribution, which is exactly why calibration is an open question. David’s framing at ICMS: Bayes needs three things to be right, the model (A1), the prior (A2) and the computation (A3); predictive Bayes drops A1 and A2 by specifying only the one-step-ahead rule, and the price is that nobody has said what its uncertainty quantification means. That price is the question of this talk. If asked why ‘martingale’: in Fong, Holmes and Walker (2023) the predictive update satisfies E[F_n(x) | 𝓕_n] = F(X_{n+1} ≤ x | x_{1:n}), so the predictive probability of any fixed set is a bounded martingale along the path and converges almost surely by Doob’s theorem; that limit is F_∞. The paper only assumes the limit exists (A1) and uses the martingale condition for one thing, to identify the centre as ξ̄n = ∫ψ dF{n:n+1}.

Predictive resampling: simulate the missing future, re-evaluate the functional

Algorithm 1 (one path; repeat M times)

Initialize the engine \mathsf F at the observed data x_{1:n}.
For i=n+1,\dots,N: draw Z_i\sim\mathsf F_{i-1:i} and update \mathsf F_{i:i+1}\leftarrow\{\mathsf F_{i-1:i},Z_i\}.
Form the empirical measure \mathbb P_{n,N} of \{x_{1:n},Z_{n+1:N}\}.
Record \vartheta^{(j)}=T(\mathbb P_{n,N}).

\Pi_N(A\mid x_{1:n}):=\Pr\{T(\mathbb P_{n,N})\in A\mid x_{1:n}\}, the truncated martingale posterior: the M draws are i.i.d. from it; with N\gg n^{3/2} the simulated data dominate.

Step 1 is the only step the bagged version will change.

One path at a time: each new draw moves the running mean a little less

Engine: z_{t+1}\mid\mathcal F_t\sim\mathcal N(\mu_t,\varkappa\sigma_n^2), \;\mu_{t+1}=\mu_t+(z_{t+1}-\mu_t)/(t+1); path variance \Sigma_\star=\varkappa\sigma^2.

Three questions for the thesis, and the gap behind each

Calibrated: a set with \Pi_N(\mathcal C_{1-\alpha}\mid x_{1:n})=1-\alpha is calibrated if \liminf_n\mathsf P_0\{\vartheta_0\in\mathcal C_{1-\alpha}\}\ge1-\alpha.

RQ1 Where does the MGP concentrate, and what is its shape, when the engine may be wrong?
Gap 1: only engine-specific results (Garelli et al. 2026; Fong and Yiu 2026a, 2026b).

RQ2 What sets the width of its credible sets, and when are they calibrated?
Gap 2: classical for likelihood posteriors (Kleijn and van der Vaart 2012); a remark for predictive Bayes.

RQ3 Can calibration be restored without extra forward simulation?
Gap 3: BayesBag (Huggins and Miller 2024) and variational bagging (Fan et al. 2025) cost a posterior fit per bag.

The working paper (Wang, Fong and Frazier 2026) answers RQ1–RQ3 for i.i.d. data and forms the first two thesis chapters; the timeline slide shows where they sit. Calibration is a frequentist, repeated-sampling notion; the MGP’s uncertainty is a conditional, single-sample notion. Credible sets feed decisions on treatment effects and risk; if they understate uncertainty, the decisions inherit the error. Existence results are available under martingale or weaker conditions; what was missing is what the posterior measures once it exists, whether it is calibrated when the engine is wrong, and a remedy that does not cost a posterior fit per bag. The definition is for ϑ₀, as in the paper’s footnote. Corollary 1 delivers coverage for ϑ⋆, the engine’s own target, which equals ϑ₀ when the engine bias b of Theorem 1 is zero (the Table 1 designs); when b ≠ 0 the set is calibrated for ϑ⋆ and off-centre for ϑ₀, the ‘Centre’ point of the Limitations slide.

2 · What the martingale posterior measures

Theorems 1–2 and Corollary 1(i) of the working paper close Gaps 1 and 2

Misspecification shrinks the MGP away from the truth

Sparse regression, n=250, p=200, engine of Fong and Yiu (2026a); heteroskedastic scale \omega_m(x)=\{1+m\,\mathbb 1(|x_1|>1)\}^{1/2}, which the engine cannot match. At m=4 a nominal 95% MGP interval covers the active coefficients only 70–80% of the time (Table 2).

Theorem 1: the MGP concentrates on the pseudo-true value \vartheta_\star=T(\mathsf F_\star)

Theorem 1 (posterior concentration)

Under A1–A4, \epsilon_n\to0, N-n\gg\epsilon_n^{-2}, K large: with probability tending to one, \Pi_N\bigl\{\|\vartheta-\vartheta_0\|>b+\delta\;\big|\;x_{1:n}\bigr\}\;\le\;1/K\quad\text{for all }\delta\ge K\epsilon_n,\qquad b=\|\vartheta_\star-\vartheta_0\|.

Robustness: the engine need not reproduce \mathsf P_0, only the functional (left); otherwise the MGP settles on the engine’s value (right). Cost: N\gg n^{3/2} steps suffice.

Say. Theorem 1: the posterior concentrates within the engine bias b of the truth, at rate ε_n, and the only demand on the simulation length is (N − n)^{1/2} ε_n → ∞. The title names the target: ϑ⋆ = T(F⋆), with F⋆ = lim_n F_n the engine’s observed-data limit, assumed to exist (Section 2.2 of the paper, A1). Two consequences. Robustness: the engine need not reproduce P₀, only the functional; on the left the Gaussian engine is wrong for Gamma data, but its mean is the data mean, so the MGP concentrates on the truth. Bias: on the right the same draws are read as a median; the engine’s limit is symmetric, so ϑ⋆ = 1 while the true median is 0.84, and more data make the mistake more confident. One line for the panel: where the posterior lands is set by the algorithm’s structure, not by the data. Clicking A1–A4 jumps to the assumptions; the ↩︎ button returns here.

If asked. The figure is an illustration built for the talk with the paper’s own ingredients: the engine of Section 3.2 with κ = ½ on Ga(2, 2) data, the misspecified column of Table 1; one growing sample from n = 25 to 800, chosen so that the running mean approaches 1 from one side; 1000 paths of ⌈n^{3/2}⌉ steps each. Formally: Π_N{‖ϑ − ϑ₀‖ > b + δ | x} ≤ 1/K for δ ≥ Kε_n with ε_n → 0. A1–A4 are: existence of the path limit, a discrepancy that controls the functional, concentration of the simulated empirical measure at rate √N, stability of the engine. The backup also keeps the schematic P₀ → ϑ₀, F⋆ → ϑ⋆. The remedy for this kind of error is a better engine, not bagging. David’s ICMS example: add a drift to the predictive mean of the Gaussian engine and the limit moves.

Theorem 2: the width is set by the engine; calibration needs \Sigma_\star\ge\Sigma_0

Theorem 2 (asymptotic shape), under A1–A6

Given the data, as n,N\to\infty, \sqrt{n}\,(\vartheta-\vartheta_\star)\;\approx\;\mathcal{N}\!\left(\sqrt{n}\,\bar\xi_n,\;\Sigma_\star\right), with \bar\xi_n the engine’s centring sequence and \Sigma_\star the variance of its path fluctuation.

Corollary 1(i): asymptotic coverage, adds A8

A nominal 1-\alpha interval covers \vartheta_\star with probability 2\,\Phi\!\left(z_{1-\alpha/2}\sqrt{\Sigma_\star/\Sigma_0}\right)-1, where \Sigma_0=\lim_n\mathrm{Var}\{\sqrt{n}\,\bar\xi_n\} is the sampling variance of the centre.

Say. Theorem 2 gives the shape: given the data, √n(ϑ − ϑ⋆) is approximately normal, with a centre ξ̄ₙ and a variance Σ⋆ that is a property of the engine’s limit. Corollary 1(i) turns that into a coverage number: a nominal 1 − α interval covers with probability 2Φ(z√(Σ⋆/Σ₀)) − 1, where Σ₀ is the sampling variance of the centre, a property of repeated samples. The picture is the Gaussian example with κ = ½ at n = 100: the teal bell is the MGP, the grey bell the sampling law of its centre; the MGP’s 95% interval leaves 17% of the grey law outside, 8.3% in each tail, so coverage is 0.83. Say the instance, it is not on the slide: here Σ⋆ = κσ² and Σ₀ = σ², so Corollary 1 gives 2Φ(z_{0.975}√κ) − 1 = 0.83 at κ = ½, the number printed in the picture. Both variances shrink like 1/n, so the picture is the same at every n: more data make the MGP more confident, not better calibrated.

If asked. The paper’s decomposition: ϑ − ϑ₀ = [T(P_{n,N}) − T(F_n)] + [T(F_n) − T(F⋆)] + [T(F⋆) − T(P₀)], that is simulation fluctuation (fixed n, N → ∞), engine fluctuation with n, and the engine bias b of Theorem 1; Theorem 2 is about the first two. Under the martingale condition of Fong et al. the centre is ξ̄ₙ = ∫ψ dF_{n:n+1}, the centering sequence of Fong and Yiu. Σ⋆ and Σ₀ coincide in the canonical well-specified i.i.d. case and only under extra conditions on the engine’s centering and curvature; F⋆ = P₀ alone is not sufficient, and nothing forces them to coincide under misspecification. Table 1: 0.77–0.86 at n = 100, 200, 500. The curve on the Theorem 3 slide gives the coverage for every ratio Σ⋆/Σ₀. The paper states Corollary 1 as a whole under the conditions of Theorem 3 plus A8, for a direction c with cᵀΣ₀c > 0; part (i) uses only Theorem 2 and A8, which is how the slide presents it; the interval is for cᵀϑ⋆, the engine’s own target.

3 · The fix: start the paths from bootstrap resamples

The bagged martingale posterior: Algorithm 2, Theorem 3 and Corollary 1(ii)

Start every path from a bootstrap resample, then pool the draws

Same forward-simulation budget: B\,M_B = M, plus B initializations

Algorithm 2 (bagged predictive resampling)

Draw B bootstrap resamples x^{*(1)},\dots,x^{*(B)} from \mathbb P_n.
For each b: initialize the engine at x^{*(b)} and run Algorithm 1 for M_B paths of N-n steps.
Pool the B\times M_B terminal draws: \widehat\Pi^\dagger_N=B^{-1}\sum_{b}\Pi_N(\cdot\mid x^{*(b)}).

Target \Pi^\dagger_N=\mathbb E^*_n\big[\Pi_N(\cdot\mid x^{*})\big]; only the initialization step of Algorithm 1 changes. Extra cost: B initializations and one update state per bag (times in Section 4).

Theorem 3: bagging adds the sampling variance, so coverage is at least nominal

Theorem 3 and Corollary 1(ii)

Under A1–A7, for \vartheta^\dagger\sim\Pi^\dagger_N(\cdot\mid x_{1:n}), \sqrt{n}\,(\vartheta^\dagger-\vartheta_\star)\;\approx\;\mathcal N\!\left(\sqrt{n}\,\bar\xi_n,\;\Sigma_0+\Sigma_\star\right), so, with A8, a nominal 1-\alpha equal-tailed interval has asymptotic coverage 2\,\Phi\!\left(z_{1-\alpha/2}\sqrt{1+\Sigma_\star/\Sigma_0}\right)-1\;\ge\;1-\alpha.

Conservative: with a correctly specified engine \Sigma_\star=\Sigma_0, so the interval is \sqrt2 times as wide as needed.

Say. Compare with Theorem 2: the only change is Σ₀ added to the variance, and the guarantee is for the pseudo-true value ϑ⋆. The curve plots both corollaries against Σ⋆/Σ₀: the MGP’s coverage falls below 0.95 whenever Σ⋆ < Σ₀ and exceeds it when Σ⋆ > Σ₀; the bMGP’s never falls below 0.95. At κ = ½ the two values are 0.83 and 0.98, that is 2Φ(z_{0.975}√(3/2)) − 1 ≈ 0.98 for the bMGP, the Gaussian example of the previous slides; the curve labels both. Conservativeness is the price: “correctly specified” means the canonical case Σ⋆ = ∫ψψᵀ dF⋆ with F⋆ = P₀, where the bMGP variance is 2Σ₀; Tables 2–3 quantify it. Say only that the coverage number uses A8, the sampling law of the centre, √n ξ̄ₙ ⇒ N(0, Σ₀). The Bernstein–von Mises analogy and A9 (smooth functionals of ℙₙ) left the slide on 2026-10-26 and belong to the answers below. This closes Gap 3; Section 4 checks it in simulations and two applications.

If asked. “Isn’t the coverage assumed rather than proved?” A8 only converts the shape into a number. The content of Theorem 3 is that bagging the initialization adds exactly the centre’s sampling variance Σ₀ to the engine’s path variance Σ⋆, at no extra forward simulation; that step does not use A8. “What if the centre is not Gaussian?” Theorem 3 still gives the shape; the coverage number loses its guarantee. That is the spike-and-slab case, whose MAP initialization is not a smooth functional, so there the evidence is empirical: active coverage 0.93–0.99 in Tables 2–3. “Why can the engine not determine the law of its own centre?” The engine is a conditional object and calibration is a repeated-sampling property; some bridge between the two is unavoidable, and A8 is that bridge. Unlike the classical Bernstein–von Mises theorem there is no universal centering sequence: the centre is set by how the engine is initialized and updated from the data (Fong and Yiu 2026a, Sections 4.3–4.4), so conditions on the engine do not fix its sampling law. For i.i.d. data A9 (ξ̄ₙ a Hadamard-differentiable functional of ℙₙ, uniformly integrable) delivers A7 and A8 together. Examples, only if asked and with the paper’s own caveats: the mean, where ξ̄ₙ is linear in ℙₙ, is immediate; the quantile and the regression centre are treated in the appendices as heuristic readings, the sample-quantile form of Σ₀ when ϑ⋆ + ξ̄ₙ is asymptotically equivalent to the sample quantile, and the sandwich form when β_n is asymptotically equivalent to the working-model MLE, not as verified instances of A9. BayesBag needs the same bootstrap consistency of its centre.

4 · Evidence and its limits

Gaussian mean, sparse regression, the ACTG175 trial, quantiles, cost; then the limitations

Over 200 replicated data sets the bagged interval keeps its promise

Table 1: MGP 77–86%, bMGP 97–99.5%, at every n and under both data laws

n	MGP, \mathcal N(0,1)	bMGP, \mathcal N(0,1)	MGP, \mathrm{Ga}(2,2)	bMGP, \mathrm{Ga}(2,2)
100	0.770	0.975	0.790	0.970
200	0.855	0.995	0.815	0.975
500	0.810	0.980	0.845	0.970

Monte Carlo coverage of nominal 95% intervals for the population mean over 200 replicated data sets from each data law; engine with \varkappa=\tfrac12, 1000 paths, N-n=\lceil n^{3/2}\rceil, bMGP with B=50, M_B=20. Asymptotic values: 0.83 and 0.98.

Sparse regression: bagging brings active coverage near the nominal level

Design	MGP, m=0	bMGP, m=0	MGP, m=4	bMGP, m=4
Linear, ridge	0.905 (0.72)	0.914 (2.77)	0.785 (0.72)	0.924 (2.84)
Linear, spike-and-slab	0.835 (0.35)	0.927 (0.63)	0.702 (0.36)	0.947 (0.90)
Student-t, ridge	0.920 (0.62)	1.000 (2.02)	0.728 (0.62)	1.000 (2.13)
Student-t, spike-and-slab	0.960 (0.30)	0.993 (0.44)	0.858 (0.30)	0.988 (0.60)

Coverage (average length) of 95% intervals for the active coefficients; n=250, p=200, 80 replications (Tables 2–3); m=4 is heteroskedasticity the engine cannot match. Bagging costs length: 1.5–3.9 times longer.

bMGP matches BayesBag: 50 initializations instead of 50 posterior fits

Gaussian ridge, active coefficients	MGP	Bayes	bMGP	BayesBag
m=0: coverage (length)	0.905 (0.72)	0.905 (0.73)	0.914 (2.77)	0.915 (2.77)
m=4: coverage (length)	0.785 (0.72)	0.787 (0.73)	0.924 (2.84)	0.924 (2.84)
Per bagged data set	—	—	one initialization	one posterior fit

The ridge posterior is conjugate, so MGP agrees with Bayes, and bMGP with BayesBag (Huggins and Miller 2024), to within 0.003 in length. Under heteroskedasticity both unbagged posteriors undercover; both bagged ones return coverage near the nominal level.

Table 2, n = 250, p = 200, 80 replications, the same design as the previous slide. Ridge prior N_p(0, I_p); Bayes is the conjugate Gaussian posterior under the same prior, BayesBag averages it over the same 50 bagged data sets with 20 draws each. Inactive coefficients, m = 0 / m = 4: MGP 0.935 / 0.799, Bayes 0.937 / 0.803, bMGP and BayesBag 1.000 / 1.000. In the ridge case Bayes is closed form and therefore the cheapest of all (0.01 s, BayesBag 0.09 s, MGP 0.75 s, bMGP 25 s): the bMGP’s advantage is confined to engines whose Bayesian counterpart needs Monte Carlo, which is why the spike-and-slab design is the one timed on the cost slide. Spike-and-slab coverage for Bayes and BayesBag was not run over the 80 data sets (about 62 hours); the paper times one replication. The Student-t designs have no closed-form posterior, so no Bayesian baseline there.

ACTG175: bagging widens the 95% region around the same centre

ACTG175 HIV trial (Hammer et al. 1996), n=2139: CD4 count at 20 weeks on 12 covariates and three treatment dummies; 95% highest-density regions and marginal densities for the treatment-effect coefficients. Under the BTPE the two posteriors are the Bayesian posterior itself and BayesBag (50 MCMC fits).

Say. Fong and Yiu’s application: 2139 patients, CD4 count at 20 weeks, twelve covariates and three treatment dummies; the two treatment-effect coefficients. Three engines: Gaussian, Student-t, and the Bayesian Student-t posterior predictive. In every panel the bagged region contains the ordinary one around the same centre: the posterior means move by less than 0.01 while the region roughly doubles in area. Under the BTPE the martingale posterior is the Bayesian posterior itself, so the third panel is Bayes against BayesBag: the same widening, at fifty MCMC fits instead of fifty initializations. Corollary 1 says the larger regions are the ones to trust when the model is unlikely to be exactly right.

If asked. Engines: GPE initialized at the Gaussian MLE, TPE at the Student-t MLE from ten restarts, 100 steps with covariates resampled from the design, then the Gaussian limit; BTPE sampled with CmdStan under N(0, 10²) priors, a half-Cauchy(0, 5) scale and ν = 5. Ordinary MGP 10,000 draws; bMGP B = 50, M_B = 20. Regions: MASS::kde2d with default bandwidths, the grid-mass HPD level of Figure 3, shared axes. Areas under the GPE: 95% region 0.077 bagged against 0.038, 50% region 0.018 against 0.009; posterior SDs of (β₂, β₃) about 0.047 for the MGP and 0.065/0.068 for the bMGP. The TPE regions sit on top of the BTPE ones (backup figure actg_joint.svg). The Gaussian engine has no Bayesian counterpart in the paper.

Gamma data at q=0.75: the QMP undercovers; bagging recovers most of it

Quantile martingale posterior of Fong and Yiu (2026b) (Table 5): coverage 33–42% for the QMP at every n, 81–93% bagged and rising with n. Calibration needs the path variance to match \Sigma_0=q(1-q)/f_{\mathsf P_0}\{Q_q(\mathsf P_0)\}^{2}; the density enters squared, and global tuning does not see it.

Say. The quantile martingale posterior of Fong and Yiu, Algorithm 4, with its own tuning rule; 200 replications per cell. Under Gaussian data both methods are near nominal at every n. Under Gamma data the QMP covers its own target about a third of the time and does not improve with n; bagging lifts coverage to 81–93%, rising with n. The reason is width, not centre: the QMP’s bias shrinks from 0.24 to 0.07 while its coverage stays flat, so the interval is too narrow. Calibration needs the engine to reproduce the density of the data at the reported quantile, and the learning rate and bandwidth are global tuning devices that do not see it.

The Σ₀ formula is the sample-quantile variance, valid when ϑ⋆ + ξ̄n is asymptotically equivalent to the sample quantile; the paper offers the matching requirement, τ⋆²/f{F⋆}{Q_q(F⋆)}² = q(1−q)/f_{P₀}{Q_q(P₀)}², as a heuristic reading of Theorems 2 and 3 for the QMP (Appendix), not as a corollary. If asked. Coverage is for Q_q(F⋆), the engine’s own limit, which separates calibration from engine bias (Q_{0.75}(F⋆) ≈ 1.40 against 1.346). The Gaussian-process variant behaves the same (QMP-GP 38–45%, bagged 81–92%). The dips (0.375 to 0.325 at n = 500 for the QMP; 0.855 to 0.810 at n = 200 bagged) are within Monte Carlo error, about 0.034 and 0.025 with 200 replications; the figure shows point estimates only, as the paper’s table does. At q = 0.50 both are fine under both DGPs (Table 4). Cyclone data: QMP intervals shrink from the median to q = 0.90 while bagged ones widen; QMP 16 s, bagged QMP about one minute, DQP by MCMC 15 minutes (backup slide).

Cost: a bMGP run is about one Gibbs fit; BayesBag needs fifty

Bagging multiplies the MGP’s time by about 35 (initializations and a per-bag update state) and still lands at the cost of one MCMC fit; bagging the Bayesian posterior multiplies one fit by 50. On the cyclone data the bagged QMP is 15 times faster than one DQP chain.

Limitations

Conservative. Correctly specified engine: variance 2\Sigma_0, intervals \sqrt2 times too wide.

Centre. Bagging widens the set; it does not move \vartheta_\star to \vartheta_0, so the engine bias b of Theorem 1 stays.

Conditions. High-level assumptions, hard to verify beyond simple engines; i.i.d. data only.

High dimension. p=300>n=250: ridge coverage stays near 0.71 after bagging (Table 7).

Speed. One MCMC fit’s worth of time, 35 times the MGP’s 0.8 s.

Say these before the panel does. Centre: bagging restores the sampling spread around the engine’s own target; if the engine’s functional is biased, the bagged set is wide around the wrong point, as in the regression figure of Section 2 where both posteriors drift with m and only the bagged one stays wide enough to reach the truth. The remedy for bias is a better engine, not more bagging. The p > n case is the clearest limit: with rank(XᵀX) ≤ n < p, regularization fixes directions the likelihood does not identify, and bootstrapping does not remove that effect; bagging captures sampling variability and cannot undo shrinkage. The speed advantage is over MCMC-based bagging: in the spike-and-slab regression the MGP takes 0.8 s, the bMGP 28 s, BayesBag 23 min; in the conjugate Gaussian ridge case the ordering reverses (Bayes 0.01 s).

5 · Plan for the candidature

Two further projects and a timeline

Project 2: engines with latent structure

Setting. Mixture engines simulate a label, then update that component (Rodrı́guez, Mena, and Walker 2025); mixtures of experts add covariate-dependent weights (Jacobs et al. 1991).

Conjecture. Unobserved labels make the engine understate sampling variability, so some functionals are not calibrated.

Questions. Which functionals are calibrated, what bagging adds, how the results extend to mixtures of experts.

Until June 2027; a proposal is in preparation.

Project 3: predictive Bayes alongside the PrO posterior

PrO. The predictively oriented posterior (McLatchie et al. 2025) targets predictive performance and adapts to misspecification.

Idea. A PrO posterior supplies the engine; the bagged MGP restores calibration where needed.

Questions. What PBI gains and loses against PrO in accuracy, calibration and cost.

From mid-2027 to the final review; exploratory, as marked on the timeline.

Timeline for the three-and-a-half-year candidature

Darkest bar: completed work, the working paper on arXiv (2609.30622) with public code, Chapters 1 and 2 of the thesis. Dashed bar: exploratory work on PBI with other predictive Bayesian methods. The last six months are for writing up. Thesis chapters in order: concentration and calibration of MGPs (RQ1–2), the bagged MGP (RQ3), latent-structure engines, PBI with PrO.

Summary

What the MGP measures. Its width is the engine’s path variance \Sigma_\star; calibration needs \Sigma_\star\ge\Sigma_0, which misspecification can break.

What bagging does. Bootstrap starting points add \Sigma_0: coverage at least nominal, same simulation budget.

What comes next. Latent-structure engines, then predictive Bayes with the PrO posterior.

Wang, Fong and Frazier (2026), arXiv:2609.30622; code: github.com/hwstat/bMGP. Joint work with David Frazier and Edwin Fong.

Backup

31–33 assumptions (Theorem 1 · Theorem 2 · Theorem 3 and Corollary 1) · 34 the pseudo-true value · 35 why N\gg n^{3/2} · 36–38 regression · 39–40 quantiles and the cyclone data · 41 cost · 42 BayesBag and Fong–Yiu · 43 the Gaussian example in closed form · 44 the κ dial · 45 references

Backup: assumptions for Theorem 1 (A1–A4)

↩ back

A1 Predictive engine. \alpha_N=n/N\to0; there is a random distribution \mathsf F_n with \mathbb P_{n,N}\Rightarrow\mathsf F_n as N\to\infty, a.s. in x_{1:n}.

A2 Discrepancy. A discrepancy \mathcal D on \mathcal P(\mathcal X) and L_T>0 with (i) \mathcal D(\nu,\nu)=0; (ii) \|T(\nu)-T(\mu)\|\le L_T\,\mathcal D(\nu,\mu); (iii) \mathcal D(\alpha\nu+(1-\alpha)\mu,\zeta)\le C\{\alpha\,\mathcal D(\nu,\zeta)+(1-\alpha)\,\mathcal D(\mu,\zeta)\}.

A3 Concentration. For some p>1 and all u>0: \Pr\{\mathcal D(\mathbb P_{N-n},\mathsf F_n)>u\mid x_{1:n},\mathsf F_n\}\le C\,u^{-p}(N-n)^{-p/2}, a.s. in x_{1:n}.

A4 Stability. For positive r_n,\varsigma_n\to0: (i) \mathcal D(\mathbb P_n,\mathsf P_0)=O_p(r_n); (ii) \alpha_N\,\mathcal D(\mathbb P_n,\mathsf F_n)=o(\varsigma_n) a.s.; (iii) a non-random \mathsf F_\star with \mathcal D(\mathsf F_n,\mathsf F_\star)=O_p(r_n).

Theorem 1 sets ε_n = ς_n + M_n r_n with M_n → ∞ and M_n r_n → 0, and asks of N only that (N − n)^{1/2} ε_n → ∞. If asked whether A2 holds in the examples: for the mean functional D = W₁ works, and a trimmed W_∞ serves quantiles. Roles, if asked. A1 makes the truncated posterior well defined and lets the simulated data dominate. A2 turns convergence of measures into convergence of the functional. A3 is the simulation rate: the simulated empirical measure is within 1/√(N − n) of the engine’s limit, which is where N ≫ n^{3/2} comes from later. A4 is where misspecification enters: the engine’s observed-data limit F⋆ may differ from P₀, so nothing here assumes the engine is right. The paper verifies the list for the Gaussian mean engine; for the copula quantile engines the limit F⋆ is not established, which is why that evidence is empirical and reported against the engine’s own target.

Backup: assumptions for Theorem 2 (A5–A6)

↩ back

Notation: \mathcal F_n=\sigma(x_{1:n}), \bar{\mathsf F}_n=\mathbb E(\mathsf F_n\mid\mathcal F_n); Theorem 2 takes A1–A4 with r_n=\varsigma_n=n^{-1/2}.

A5 Smoothness of T. There is \psi:\mathcal X\to\mathbb R^{d_\vartheta} with \int\psi\,\mathrm d\mathsf F_\star=0 such that, for any H_n with \mathcal D(H_n,\mathsf F_\star)=O_p(n^{-1/2}), \sqrt n\{T(H_n)-T(\mathsf F_\star)-\int\psi\,\mathrm dH_n\}=o_p(1).

A6 Path fluctuation. For \mathcal G\ni1,\psi an \mathsf F_\star-Donsker class, there is a tight mean-zero Gaussian \mathbb G_{\mathsf F_\star} with r_n^{-1}(\mathsf F_n-\bar{\mathsf F}_n)\mid\mathcal F_n\Rightarrow\mathbb G_{\mathsf F_\star} in \ell^\infty(\mathcal G), in \mathsf P_0-probability.

Then \bar\xi_n:=\int\psi\,\mathrm d\bar{\mathsf F}_n and \Sigma_\star:=\mathrm{Var}\{\int\psi\,\mathrm d\mathbb G_{\mathsf F_\star}\} are the centre and the variance in Theorem 2.

Backup: assumptions for Theorem 3 and Corollary 1 (A7–A8)

↩ back

Notation: x^* a bootstrap resample of x_{1:n}, \mathsf F^*_n=\mathsf F_n(\cdot\mid x^*), \bar{\mathsf F}^*_n=\mathbb E[\mathsf F^*_n\mid\mathcal F^*_n], \bar\xi^*_n=\int\psi\,\mathrm d\bar{\mathsf F}^*_n, \Sigma_0:=\lim_n\mathrm{Var}_{\mathsf P_0}\{\sqrt n\,\bar\xi_n\}.

A7 Bootstrap centre. Under the bootstrap law, \sqrt n(\bar\xi^*_n-\bar\xi_n)\mid\mathcal F_n\Rightarrow\mathcal N(0,\Sigma_0) in \mathsf P_0-probability.

A8 Centre (Corollary 1 only). \sqrt n\,\bar\xi_n\Rightarrow\mathcal N(0,\Sigma_0).

A9 (sufficient for A7–A8). I.i.d. data, \mathcal G \mathsf P_0-Donsker, \bar\xi_n=\Xi(\mathbb P_n) with \Xi(\mathsf P_0)=0 and \Xi Hadamard differentiable at \mathsf P_0, \{n\,\bar\xi_n\bar\xi_n^\top\} uniformly integrable.

Theorem 3 uses A1–A7 at the rate n^{-1/2}, with A3, A4 and A6 also holding for the resampled quantities. Under A9, Σ₀ = Var{Ξ̇{P₀}(G{P₀})} by the functional delta method and bootstrap consistency (van der Vaart and Wellner 1996). A7 and A8 state the same limit for different objects, the bootstrap law of the centre and its sampling law; without the structure ξ̄ₙ = Ξ(ℙₙ) the two limits may differ, which is why they are kept apart. Roles, if asked. A7 says the bootstrap reproduces the sampling variability of the centre, which is exactly what bagging injects. A8 is needed only to turn the Gaussian shape into a coverage number. Unlike the classical Bernstein–von Mises theorem there is no universal centering sequence for MGPs: conditions on the engine do not pin down the sampling law of ξ̄ₙ, so A8 is a separate assumption; for i.i.d. data A9 delivers both.

Backup: the pseudo-true value, schematically

The engine’s observed-data limit \mathsf F_\star need not equal \mathsf P_0; Theorem 1 says the MGP concentrates on \vartheta_\star=T(\mathsf F_\star), within b=\|\vartheta_\star-\vartheta_0\| of the truth, and b=0 whenever the engine reproduces the functional.

Backup: why N\gg n^{3/2}

\mathbb P_{n,N}=\alpha_N\mathbb P_n+(1-\alpha_N)\mathbb P_{N-n} with \alpha_N=n/N.

Concentration needs only (N-n)^{1/2}\epsilon_n\to\infty, which for \epsilon_n\asymp n^{-1/2} is implied by n/N\to0.

The Gaussian shape needs the observed data to be negligible at the \sqrt n scale: \sqrt n\,\alpha_N=n^{3/2}/N\to0, hence N\gg n^{3/2}. In the experiments N-n=\lceil n^{3/2}\rceil for the mean and quantile designs.

Backup: the regression coverage as a picture (Tables 2–3)

Active coefficients, 80 replications. Under homoskedastic errors the MGP is close to nominal and bagging is conservative; under heteroskedastic errors the MGP undercovers and bagging restores coverage.

Backup: regression results in full (Tables 2–3, n=250, p=200)

Example	Method	Active cov.	Active len.	Inactive cov.	Inactive len.
Lin. ridge, m=0	MGP / bMGP	0.905 / 0.914	0.724 / 2.772	0.935 / 1.000	0.722 / 2.747
Lin. spike, m=0	MGP / bMGP	0.835 / 0.927	0.352 / 0.630	1.000 / 1.000	0.071 / 0.082
Lin. ridge, m=4	MGP / bMGP	0.785 / 0.924	0.724 / 2.844	0.799 / 1.000	0.722 / 2.820
Lin. spike, m=4	MGP / bMGP	0.702 / 0.947	0.355 / 0.895	0.987 / 1.000	0.075 / 0.238
t ridge, m=0	MGP / bMGP	0.920 / 1.000	0.619 / 2.015	0.915 / 1.000	0.618 / 2.000
t spike, m=0	MGP / bMGP	0.960 / 0.993	0.300 / 0.439	0.999 / 1.000	0.076 / 0.092
t ridge, m=4	MGP / bMGP	0.728 / 1.000	0.619 / 2.126	0.760 / 1.000	0.618 / 2.105
t spike, m=4	MGP / bMGP	0.858 / 0.988	0.300 / 0.597	0.997 / 1.000	0.076 / 0.152

80 replications; 50 active coefficients in the linear design, 5 in the Student-t design. Bayes and BayesBag agree with MGP and bMGP to within 0.003 in length in the ridge case.

Backup: when p>n (p=300, n=250, Tables 7–8)

Linear spike-and-slab. Active coverage rises from 0.828 to 0.909 (m=0) and from 0.677 to 0.917 (m=4).

Linear ridge. Active coverage stays near 0.71 after bagging although intervals almost double: with \mathrm{rank}(X^\top X)\le n<p, regularization fixes directions the likelihood does not identify, and bootstrapping does not remove that effect.

Student-t. Ridge is already conservative; spike-and-slab rises from 0.880 to 0.993 under m=4.

Bagging is not a universal correction: it adds sampling variability, nothing else.

Backup: quantiles at q=0.50 and in the tail

q=0.50 (Table 4). QMP coverage 0.975–0.995 under both DGPs; bagged 0.990–1.000.

Target. Coverage is for Q_q(\mathsf F_\star), the engine’s own limit, estimated from an oracle sample of 2\times10^4 draws (stable up to 10^5); under \mathrm{Ga}(2,2), Q_{0.75}(\mathsf F_\star)\approx1.40 against Q_{0.75}(\mathsf P_0)=1.346, so engine bias and calibration are separated.

Tail. At q=0.95 the oracle target does not settle under Gamma data: the initial quantile curve follows the sample maximum, which grows like \tfrac12\log n; the tail experiment is therefore assessed against the population quantile.

Backup: cyclone wind speeds, bagged intervals widen in the upper tail

q	Method	Mean (SD)	95% interval	Length	Draws / ESS
0.50	QMP	0.447 (0.288)	[−0.114, 1.017]	1.131	10,000
	Bagged QMP	0.441 (0.393)	[−0.338, 1.254]	1.591	1,000
	DQP (MCMC)	0.343 (0.174)	[0.071, 0.677]	0.606	4
0.90	QMP	1.036 (0.171)	[0.700, 1.372]	0.672	10,000
	Bagged QMP	1.061 (0.436)	[0.226, 1.930]	1.704	1,000
	DQP (MCMC)	1.426 (0.269)	[0.844, 1.933]	1.089	10

Slope in year of the conditional q-quantile of lifetime maximum wind speed, 291 North Atlantic cyclones, 1981–2006 (Elsner, Kossin, and Jagger 2008). QMP intervals shrink from the median to q=0.90; bagged ones widen. Run times: QMP 16 s, bagged QMP about 1 min, DQP (An and MacEachern 2024) 15 min.

Backup: computational cost (Table 9 and the cyclone run times)

Procedure	One replication
Gibbs sampler, one posterior fit	28.5 s
BayesBag, 50 Gibbs fits	23.2 min
MGP, 1 initialization and 1000 paths	0.79 s
bMGP, 50 initializations and 1000 paths	28.0 s

Linear spike-and-slab design, n=250, p=200, one core. The 50 EM initializations take 3.6 s; the rest is the predictive update, repeated per bag because each has its own covariance state. In the conjugate ridge case the ordering reverses (Bayes 0.01 s, bMGP 25 s): the advantage is confined to engines whose Bayesian counterpart needs Monte Carlo.

Cyclone data: QMP 16 s, bagged QMP 59 s, QMP-GP 1.1 s, bagged QMP-GP 20 s, DQP 15 min.

Backup: relation to bagged posteriors and to Fong and Yiu (2026)

BayesBag (Huggins and Miller 2024) and variational bagging (Fan et al. 2025) also average posteriors over bootstrap resamples; both need a posterior computation per bag. The bMGP re-initializes an engine per bag and keeps predictive resampling, so the forward-simulation budget is unchanged.

Fong and Yiu (2026) give asymptotics for a parametric martingale posterior with a specific engine and centering sequence (their Theorem 3). Theorems 1–2 here are engine agnostic and functional based; the calibration analysis and the bagging remedy are new.

Existence of the limiting MGP is taken as given (Fong, Holmes, and Walker 2023; Battiston and Cappello 2025); the paper is about what the posterior measures once it exists.

Backup: the Gaussian example in closed form

Engine z_{t+1}\mid\mathcal F_t\sim\mathcal N(\mu_t,\varkappa\sigma_n^2) with \mu_{t+1}=\mu_t+(z_{t+1}-\mu_t)/(t+1). The path \mu_t is a martingale with increment variances \varkappa\sigma_n^2/(t+1)^2, so n\,\mathrm{Var}\{T(\mathbb P_{n,N})\mid x_{1:n}\}\to\varkappa\sigma^2=\Sigma_\star,\qquad \Sigma_0=\sigma^2 . At \varkappa=\tfrac12: MGP coverage 2\Phi(z_{0.975}/\sqrt2)-1\approx0.83; bMGP coverage 2\Phi(z_{0.975}\sqrt{3/2})-1\approx0.98. Table 1: 0.77–0.86 and 0.97–0.995 at n\in\{100,200,500\}.

Backup: the κ dial moves the MGP, not the sampling variability

References

An, Hyoin, and Steven N. MacEachern. 2024. “A Process of Dependent Quantile Pyramids.” Journal of Nonparametric Statistics, 1–25. https://doi.org/10.1080/10485252.2024.2305811.

Battiston, Marco, and Lorenzo Cappello. 2025. “Bayesian Predictive Inference Beyond Martingales.” arXiv Preprint arXiv:2507.21874.

Berti, Patrizia, Emanuela Dreassi, Luca Pratelli, and Pietro Rigo. 2021. “A Class of Models for Bayesian Predictive Inference.” Bernoulli 27 (1): 702–26.

Elsner, James B., James P. Kossin, and Thomas H. Jagger. 2008. “The Increasing Intensity of the Strongest Tropical Cyclones.” Nature 455 (7209): 92–95.

Fan, S, I Ohn, D Dunson, and L Lin. 2025. “Variational Bagging: A Robust Approach for Bayesian Uncertainty Quantification.” arXiv Preprint arXiv:2511.20594.

Fong, Edwin, Chris Holmes, and Stephen G Walker. 2023. “Martingale Posterior Distributions.” Journal of the Royal Statistical Society Series B: Statistical Methodology 85 (5): 1357–91.

Fong, Edwin, and Andrew Yiu. 2026a. “Asymptotics for a Class of Parametric Martingale Posteriors.” Biometrika 113 (2): asag007.

———. 2026b. “Bayesian Quantile Estimation and Regression with Martingale Posteriors.” Journal of the Royal Statistical Society Series B: Statistical Methodology 88 (4): 1130–59. https://doi.org/10.1093/jrsssb/qkaf080.

Fortini, Sandra, and Sonia Petrone. 2025. “Exchangeability, Prediction and Predictive Modeling in Bayesian Statistics.” Statistical Science 40 (1): 40–67.

Garelli, Samuele, Fabrizio Leisen, Luca Pratelli, and Pietro Rigo. 2026. “Asymptotics of Predictive Distributions Driven by Sample Means and Variances.” Journal of Applied Probability.

Hammer, Scott M., David A. Katzenstein, Michael D. Hughes, Holly Gundacker, Robert T. Schooley, Richard H. Haubrich, W. Keith Henry, et al. 1996. “A Trial Comparing Nucleoside Monotherapy with Combination Therapy in HIV-Infected Adults with CD4 Cell Counts from 200 to 500 Per Cubic Millimeter.” New England Journal of Medicine 335 (15): 1081–90.

Huggins, Jonathan H, and Jeffrey W Miller. 2024. “Reproducible Parameter Inference Using Bagged Posteriors.” Electronic Journal of Statistics 18 (1): 1549–85. https://doi.org/10.1214/24-EJS2237.

Jacobs, Robert A., Michael I. Jordan, Steven J. Nowlan, and Geoffrey E. Hinton. 1991. “Adaptive Mixtures of Local Experts.” Neural Computation 3 (1): 79–87.

Kleijn, B. J. K., and A. W. van der Vaart. 2012. “The Bernstein-Von-Mises Theorem Under Misspecification.” Electronic Journal of Statistics 6: 354–81. https://doi.org/10.1214/12-EJS675.

Lee, Hyungi, Eunggu Yun, Giung Nam, Edwin Fong, and Juho Lee. 2023. “Martingale Posterior Neural Processes.” In International Conference on Learning Representations.

McLatchie, Yann, Badr-Eddine Chérief-Abdellatif, David T. Frazier, and Jeremias Knoblauch. 2025. “Predictively Oriented Posteriors.” arXiv preprint arXiv:2510.01915.

Ng, Kenyon, Edwin Fong, David T. Frazier, Jeremias Knoblauch, and Susan Wei. 2026. “TabMGP: Martingale Posterior with TabPFN.” In International Conference on Machine Learning.

Rodrı́guez, Carlos E., Ramsés H. Mena, and Stephen G. Walker. 2025. “Martingale Posterior Inference for Finite Mixture Models and Clustering.” Journal of Computational and Graphical Statistics 34 (4): 1253–62.