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Time-partitioning Gaussian Process Mixture Model

Guilherme Piantino, Hedibert F. Lopes

arXiv:2610.20860v1stat.MLcs.LG

TL;DR

Macro-financial time series can shift between regimes, but their number, timing, and duration are often unknown. TPGPMM combines an adaptive prior on contiguous segmentations with regime-specific Gaussian processes. Its fixed-noise specification favors sparse, event-aligned segmentation, while estimating noise improves fit and forecasts but can reduce segmentation sparsity or event alignment.

  • Problem

    The number, timing, and duration of regimes in macro-financial time series are typically unknown, yet identifying them matters for risk management, forecasting, and policy analysis.

    Source: §1 · p. 1

  • Method

    TPGPMM infers contiguous regimes under a Pitman–Yor prior and models each regime with a Gaussian process.

    Source: §3 · p. 3, §9 · p. 17

  • Results

    Fixing noise at a conservative level keeps TPGPMM’s segmentation sparse and event-aligned, while estimating it improves fit and forecasts but can fragment the segmentation or miss events.

    Source: §9 · p. 18

  • Takeaways & Limitations

    The model’s regime-identification and prediction aims trade segmentation sparsity and event alignment against predictive accuracy.

    Source: §9 · p. 18

    TPGPMM is best suited to persistent regimes with non-Markovian within-regime dynamics and is less suited to rapidly switching or recurring states.

    Source: §9 · p. 18

Abstract

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We propose the Time-partitioning Gaussian Process Mixture Model (TPGPMM), a Bayesian nonparametric framework for identifying persistent regimes and modelling nonstationary dependence in time series. The model combines an adaptive Pitman--Yor prior on contiguous segmentations with regime-specific Gaussian processes, without imposing Markovian within-regime dependence. Learning the Pitman--Yor discount parameter allows adaptation to homogeneous and heterogeneous regime lengths. A split--merge Metropolis-within-Gibbs sampler with exact Gibbs boundary updates jointly learns change points and model hyperparameters, quantifying uncertainty over segmentations and within-regime dynamics. With an RBF kernel and a fixed, conservative observation-noise level, TPGPMM favours sparse segmentations that distinguish persistent shifts from transient shocks. Simulations demonstrate recovery of the regime structure. Applications to the CBOE Volatility Index (VIX) and the 10-year United States Treasury yield identify regimes aligned with major market and monetary-policy events. Variants that estimate observation noise improve predictive accuracy relative to the main specification, but can produce less sparse or less event-aligned segmentations. A variant with regime-specific means fits better than a fitted stationary Gaussian process and forecasts about as accurately. These results highlight a trade-off between interpretable regime identification and predictive accuracy.

1 Introduction

The paper introduces TPGPMM to identify contiguous regimes and model nonstationary dependence without imposing Markovian within-regime dynamics. Its two aims—regime identification and prediction—favor different observation-noise choices.

1 Introduction

  • TPGPMM infers the number and locations of regimes through a piecewise GP structure for time-ordered data.It is designed to capture multiple regime changes without imposing a rigid Markovian structure.Source: §1 · p. 2
  • Regime identification uses posterior uncertainty over contiguous partitions to distinguish persistent shifts from transient shocks.For prediction, posterior-averaged regime-specific GPs yield a nonstationary predictive distribution within and beyond the observed series.Source: §1 · p. 2
  • TPGPMM differs from GP expert mixtures by targeting contiguous regimes, and from hazard-based change-point GP methods by using a position-neutral partition prior and batch inference.Candidate boundaries are evaluated using global posterior evidence.Source: §1 · p. 2
  • Estimating observation noise improves predictive fit and forecasts relative to the main specification, but can yield less sparse or less event-aligned segmentations.With a regime mean, the model fits better than a fitted stationary GP and forecasts about as accurately.Source: §1 · p. 3

2 The model

The model represents a time-ordered series as contiguous blocks, each with its own GP hyperparameters and latent function. The partition determines the blocks, while the observations depend on the latent evaluations.

2 The model

  • The time-ordered observations are divided into contiguous blocks, each interpreted as a distinct regime.The partition is represented by a composition of block lengths, with a minimum regime length of three observations.Source: §2 · p. 3
  • Regime-specific hyperparameters are piecewise constant over time and change only at internal boundaries.Conditional on the partition, block hyperparameters are independent and identically distributed.Source: §2 · p. 3
  • Each block has a latent function governed by its GP hyperparameters, and observations are conditionally independent given the latent values.Each observation depends only on its corresponding latent evaluation.Source: §2 · p. 3
  • The partition defines the contiguous segments, while regime-specific hyperparameters govern non-Markovian GP dependence within each segment.The graphical representation separates the global partition from local parameters and depicts within-regime covariance among latent nodes.Source: §2 · p. 3

3 Prior on the segmentation

The segmentation prior uses a Pitman–Yor process mapped to ordered compositions, allowing the number and lengths of contiguous regimes to be inferred. Learning its discount parameter adapts the prior to different regime-length patterns.

3 Prior on the segmentation

  • The Pitman–Yor prior lets the data determine the number and locations of regimes rather than fixing the number in advance.Its partition probability depends on block sizes and the number of blocks.Source: §3 · p. 3
  • Mapping the exchangeable partition to ordered compositions preserves the marginal distribution of K and makes block orders equally likely conditional on their sizes.This position neutrality leaves the prior to govern regime number and lengths, while the likelihood drives change placement.Source: §3 · pp. 3–4
  • The prior is conditioned on a minimum regime length, and its normalising constant must be included when updating prior hyperparameters.The minimum length is three observations in the model description.Source: §3 · p. 4
  • Estimating the Pitman–Yor discount σ lets the data adapt between homogeneous lengths and a heavy-tailed mix of short and long regimes.The Dirichlet process is the limiting case as σ approaches zero.Source: §3 · p. 4

4 Gaussian process likelihood within blocks

Within each contiguous block, observations are modelled using an independent GP. The main specification uses a shared, fixed conservative noise level and an RBF kernel, with alternative specifications used to assess noise estimation and within-regime model choices.

4 Gaussian process likelihood within blocks

  • Each regime is modelled with an independent GP, and the block likelihood uses its covariance matrix plus observation-noise variance.For stationary kernels on equally spaced inputs, the covariance matrix is symmetric Toeplitz.Source: §4 · pp. 4–5
  • Observation-noise variance is shared across regimes so that regimes differ through their GP hyperparameters.This choice is intended to prevent regime shifts from being explained as noise changes.Source: §4 · p. 5
  • The RBF kernel induces smooth latent functions without parametric Markov transitions, while alternatives estimate noise and vary kernel smoothness or add a regime mean.The alternatives include a Matérn-3/2 kernel and a Matérn-3/2 model with an integrated mean for each regime.Source: §4 · p. 5
  • The main specification fixes noise conservatively to help distinguish observation noise from within-regime variation and discourage spurious change points.Fixing noise also reduces ambiguity in the length-scale and amplitude parameters.Source: §4 · p. 5

5 Posterior Computation

Posterior inference jointly samples the segmentation and model parameters, using split–merge moves and exact boundary updates. Posterior summaries describe boundary uncertainty, regime assignments, and Pitman–Yor parameters.

5 Posterior Computation

  • Inference jointly samples the segmentation and model parameters because the marginal posterior of ρ is analytically intractable.When observation noise is estimated, its parameter and prior are included.Source: §5 · p. 5

5.1 Inference Algorithm

  • Split–merge reversible-jump moves change the partition globally, with auxiliary variables making splits dimension matched to their exact inverse merges.The sampler also updates two boundaries from their full conditionals and kernel parameters for three randomly chosen blocks.Source: §5.1 · p. 6
  • Toeplitz covariance structure lets one factorisation evaluate every block prefix and suffix, enabling exact boundary draws and informed split-point proposals.Split points are proposed in proportion to the target given the parent block’s hyperparameters.Source: §5.1 · p. 6

5.2 Posterior Summaries

  • Posterior summaries include boundary probabilities, probabilities of a boundary within ±10 trading days, and a similarity matrix for shared regime assignments.The posterior of K is summarized empirically, while σ and θ are reported using means, medians, and equal-tailed 95% credible intervals.Source: §5.2 · pp. 6–7

6 Experimental setup

The experiments assess regime identification and nonstationary prediction using simulated data and two macro-financial series. They compare TPGPMM specifications with change-point and GP-based alternatives.

6 Experimental setup

  • Regime identification is assessed on simulated data with known change-points and on two macro-financial series compared with documented events.Nonstationary prediction is assessed in-sample and beyond the series’ ends.Source: §6 · p. 7
  • The empirical series are daily log-VIX and standardized 10-year US Treasury yields from January 2015 to December 2023, with 2024 trading days held out for forecasting.The series contain 2,264 and 2,251 observations, respectively; fixed noise σ_n = 0.1 corresponds to about 9 basis points of yield.Source: §6 · p. 7
  • Regime identification comparisons include BOCPD, HDP-SLDS, and Martinez–Mena models with independent-observation and autoregressive specifications.The Martinez–Mena Model B fixes autocorrelation at zero, whereas Model A allows Markovian autoregressive dependence.Source: §6 · p. 7
  • Prediction comparisons include stationary RBF and Matérn-3/2 GPs, laGP, HDP-SLDS, and Martinez–Mena predictive distributions.Stationary GP hyperparameters are fitted by maximizing marginal likelihood.Source: §6 · p. 7
  • The main specification uses an RBF kernel and fixed σ_n = 0.1; three prediction variants estimate noise, with two also using Matérn-3/2 and one adding a regime mean.The fixed noise value is deliberately conservative and is about twice short-horizon estimates for log-VIX and two to three times estimates for yields.Source: §6 · p. 7
  • Empirical fits use four chains per series, initialized with 1, 5, 15, or 30 equal blocks, with 60,000 VIX and 20,000 Treasury iterations.The first half is burn-in and every fifth draw is retained; a 60,000-iteration VIX chain takes about 8 minutes for TPGPMM versus up to 50 minutes for alternatives on one Apple Silicon core.Source: §6 · pp. 7–8

7 Regime identification

This section evaluates TPGPMM’s change-point recovery in simulations and examines inferred regimes in VIX and Treasury-yield data. Its results show strong simulation recovery and sparse VIX segmentation, while event-boundary timing varies in the yield analysis.

7.1 Simulation Study

  • In power-law simulations, TPGPMM achieves F1 = 0.996 and CP_MAE = 0.05, recovering K = 10 in 92% of replications; homogeneous F1 is 0.981, slightly below MM-A’s 0.985.In the homogeneous scenario, TPGPMM slightly over-segments (K̂ = 10.3 ± 0.7; exact K in 72% of replications), but its CP_MAE is lower than every baseline.Source: Table 1 · p. 8, §7.1 · p. 8
  • Across specifications, posterior σ separates the scenarios: homogeneous seeds yield 0.104–0.148, while power-law seeds yield 0.516–0.540.The separation is independent of kernel, noise model, and regime mean, indicating adaptation to segment-size distribution through the partition prior.Source: §7.1 · p. 9, Figure 2 · p. 9

7.2 VIX

  • TPGPMM matches China, Volmageddon, and COVID-19 among the six VIX benchmark events, but assigns no boundary mass to Brexit, the Russia–Ukraine invasion, or 2023 bank turmoil.Those short-lived spikes are absorbed within regimes that began earlier, so the method targets persistent structural shifts rather than every major event.Source: §7.2 · p. 11
  • For VIX, near-unit-root MM-A merges China and COVID-19 into long regimes, whereas TPGPMM’s non-Markovian GP can absorb transient spikes within blocks, supporting sparse segmentation.The authors attribute this difference to the within-block likelihood: adding GP hyperparameters is penalized unless persistence and amplitude differ across a candidate boundary.Source: §7.2 · p. 11
  • On VIX, only TPGPMM produces a segmentation described as sparse, confident, and aligned with persistent episodes; its boundary probability exceeds 0.5 at 7 locations.BOCPD and HDP-SLDS produce denser, more diffuse probabilities, with 20 and 38 locations above 0.2, respectively.Source: §7.2 · pp. 10–11, Figure 3 · p. 10

7.2.1 Interpretation of the inferred regimes

  • The VIX modal partition places the volatile China-sell-off phase in block 3, which absorbs Brexit, and groups the exceptionally calm November 2016–January 2018 period into blocks 4–6.Block 7 begins with Volmageddon and absorbs the late-2018 sell-off; block 8 is the COVID-19 regime.Source: §7.2.1 · pp. 12–13

7.3 10-year Treasury yield

  • The Treasury-yield MAP partition has five regimes: the COVID-and-quantitative-easing boundary is certain, while 2019 easing and 2022 tightening are detected with uncertain timing.The 2019 boundary has probability 0.22 within ±10 days of the modal date, with more posterior mass (0.69) in early September; the late-February 2022 boundary has probability 0.47.Source: §7.3 · pp. 13–14

8 Nonstationary prediction

The posterior predictive mixture carries uncertainty across partitions and GP parameters, while results reveal a trade-off: estimated-noise variants predict better, but can alter or fragment regime segmentations.

8 Nonstationary prediction

  • The regime-mean alternative forecasts about as accurately as a fitted stationary GP, while estimating noise improves prediction but can fragment or change inferred segmentations.On VIX, estimated-noise alternatives have CRPS/sd values of 0.093, 0.068, and 0.046, versus 0.122 for TPGPMM; the regime-mean alternative uses 51 regimes on VIX and 13 on the Treasury yield.Source: §8 · p. 14, §8.3 · p. 17

8.1 Posterior predictive distribution

  • Averaging conditional GP predictions across posterior draws propagates uncertainty in partitions and GP parameters; forecasts average each draw’s prediction from its final regime.The posterior predictive mixture also widens bands in regions of structural ambiguity.Source: §8.1 · pp. 14–15

8.2 In-sample fit

  • On log-VIX, TPGPMM’s RMSE/sd is 0.208 and CRPS/sd is 0.122, less close than fitted stationary GPs but about as close as MM-A.The regime-mean alternative fits best among the alternatives, with RMSE/sd 0.076 and CRPS/sd 0.046.Source: §8.2 · pp. 15–16, Table 5 · p. 16
  • On the Treasury yield, TPGPMM’s RMSE/sd is 0.061 and CRPS/sd is 0.037, while the regime-mean alternative fits best with RMSE/sd 0.021 and CRPS/sd 0.013.Fitted stationary GPs have CRPS/sd 0.025 for RBF and 0.018 for Matérn-3/2.Source: §8.2 · p. 16, Table 6 · p. 16

8.3 Out-of-sample forecasting

  • On log-VIX, the regime-mean alternative has the best CRPS at every horizon; at one day, fitted Matérn has slightly lower RMSE, 0.470 against 0.476.The five-day differences are small: RMSE/sd is 0.940 against 0.946, and CRPS/sd is 0.482 against 0.496.Source: §8.3 · pp. 16–17, Table 7 · p. 17
  • On the Treasury yield, fitted Matérn has better CRPS at every horizon, while regime-mean and fitted Matérn RMSE are close.At five days, CRPS is 0.347 for fitted Matérn against 0.377 for regime-mean; the regime-mean alternative’s five-day predictive bands are too wide.Source: §8.3 · p. 17

9 Conclusions

TPGPMM jointly infers contiguous regimes and regime-specific Gaussian processes, balancing interpretable segmentation against predictive accuracy. Its conservative fixed noise supports sparse, event-aligned segmentation, while estimating noise improves prediction but may fragment regimes or miss events.

9 Conclusions

  • TPGPMM infers a contiguous segmentation jointly with regime-specific Gaussian processes and learns partition and kernel hyperparameters, including the Pitman–Yor discount.The main specification fixes observation noise at a conservative, data-informed level.Source: §9 · pp. 17–18
  • The conservative fixed noise keeps segmentation sparse and event-aligned, whereas estimating noise improves fit and forecasts but can fragment segmentations or miss events.The regime-mean alternative fits better than a fitted stationary GP and forecasts about as accurately.Source: §9 · p. 18
  • Segmentation reflects three ingredients: an adaptive Pitman–Yor prior, a non-Markovian GP likelihood that absorbs short-range fluctuations, and conservative noise that discourages boundaries.The text contrasts these with estimated-noise fragmentation, Markovian regimes, independent within-block observations, and diffuse frequent boundaries.Source: §9 · p. 18
  • TPGPMM best suits persistent regimes with non-Markovian within-regime dynamics; its forecasts cannot anticipate a new regime, and online use requires a sequential composition prior.Inference is cubic in regime lengths, making fewer and longer regimes slower; the approach is practical for series of several thousand observations.Source: §9 · p. 18

A Derivation of the Pitman–Yor prior on ordered segmentations

The derivation maps the Pitman–Yor exchangeable partition law to ordered compositions and hence to contiguous segmentations. Its discount parameter controls block-count growth, while the resulting prior is defined for a fixed series length.

A Derivation of the Pitman–Yor prior on ordered segmentations

  • For σ = 0, the Pitman–Yor process reduces to the Dirichlet process, which implies logarithmic growth in the number of blocks; the broader Gibbs-type family allows polynomial growth.The discount parameter provides explicit control over block-count growth.Source: A Derivation of the Pitman–Yor prior… · p. 22
  • The ordered composition probability is obtained by summing over ordered label assignments for each block-size sequence, using the multinomial count of those assignments.The construction uses the exchangeable partition probability function and assigns the expression in (4) to compositions.Source: A Derivation of the Pitman–Yor prior… · pp. 22–23
  • The Pitman–Yor prior on ordered compositions transfers directly to contiguous segmentations, preserving the law on the number and sizes of blocks while restricting support to temporally ordered regimes.Each composition determines a unique contiguous segmentation.Source: A Derivation of the Pitman–Yor prior… · pp. 22–23
  • The prior is not projective in series length, so removing the last observation does not recover the prior for the shorter series; the model is therefore defined for a fixed series length.For n = 5, 8, and 11, the reported total-variation distance is 0.06–0.10; an online version needs a different construction.Source: A Derivation of the Pitman–Yor prior… · p. 23

B Specifications of the within-regime model

The within-regime specifications share the partition prior and most hyperpriors, but vary the kernel, regime means, and treatment of observation noise. These choices alter the model’s within-regime flexibility and noise treatment.

B Specifications of the within-regime model

  • The four within-regime specifications share the partition prior, minimum regime length, priors on σ and θ, and length-scale prior, but differ in kernel, regime means, and whether observation noise is fixed or estimated.The alternatives are summarized in Table 9.Source: B Specifications of the within-regime… · p. 23, Table 9 · p. 23
  • The estimated-noise specification uses one noise level shared across regimes, and each noise update recomputes all block likelihoods because the noise variance enters every block covariance.Its prior is centered on the value fixed in the main specification.Source: B Specifications of the within-regime… · p. 23
  • The Matérn-3/2 alternative replaces the RBF kernel, allowing rougher within-regime variation while retaining stationary, symmetric Toeplitz block covariances.Its sample paths are continuous and once mean-square differentiable, unlike the infinitely differentiable RBF paths.Source: B Specifications of the within-regime… · p. 24
  • The regime-mean alternative adds an independent constant level to each regime’s zero-mean Matérn-3/2 GP and integrates that mean into the block marginal likelihood.The mean and variance offsets are set to the series sample mean and variance.Source: B Specifications of the within-regime… · p. 24

C MCMC implementation details

This section documents the computational and evaluation procedures for TPGPMM, covering sampler updates and checks, comparator inference and benchmarks, evaluation metrics and summaries, and the simulation-study configuration.

C.1 Block A: structural updates of ρ

  • Structural updates enforce a minimum block length of 3, use reversible split–merge proposals, and follow each move with exact Gibbs draws of two regime boundaries.The split proposal includes a uniform component so every allowed split point remains reachable.Source: §C.1 · p. 24, §D.2 · p. 27

C.2 Block B: kernel hyperparameter update

  • Kernel parameters use random walks on log scales, while Pitman–Yor parameters use logit and log scales; estimated noise updates recompute every block likelihood.The kernel proposal scale is adapted toward 35% acceptance during burn-in and then fixed; using a natural-scale ratio for log-scale kernel updates shifts the prior, by −0.36 for log ℓk and −0.64 for log s2 k.Source: §C.2 · p. 25, §C.3 · p. 25, §D.4 · p. 28

C.4 Block D: bookkeeping

  • The bookkeeping recomputes block log-marginals every 200 iterations, discards half of each chain as burn-in, and thins retained draws every five iterations.Prior-only checks with all terms included yield total-variation distance 0.014 from the exact law of K; with the likelihood on, an exact grid sampler agrees on posterior K and boundary probabilities.Source: §C.4 · p. 26, §D.7 · p. 29

D.2 Split and merge

  • A valid split replaces two log hyperparameters with four using exactly two auxiliary variables and an invertible map; omitting normalized prior constants instead targets a distribution proportional to 3.02^K times the intended target.The reported factor uses sℓ = 0.6 and ss = 0.8; independently perturbing each child uses four variables and does not define inverse split and merge maps.Source: §D.2 · pp. 26–27

D.3 Boundary moves

  • Exact Gibbs boundary draws sample the full conditional without rejection, and two block factorizations evaluate all candidate positions at about the cost of one shuffle proposal.The conditional has at most T possible boundary positions; a uniform shuffle instead proposes mostly negligible positions for long blocks.Source: §D.3 · pp. 27–28

D.5 Pitman–Yor parameters and the truncation constant

  • Omitting the truncation constant Zn from discount and strength updates changes the target hyperprior and largely masks adaptation of σ to segment sizes.The tilted prior means are σ ≈ 0.04 and θ ≈ 2.2 rather than 0.5 and 10; in simulations, posterior mean σ falls from 0.121 and 0.526 to about 0.03 and 0.06.Source: §D.5 · p. 28

E.1 BOCPD: Bayesian online change-point detection with a Normal-Inverse-Gamma conjugate underlying predictive model

  • BOCPD uses a Normal–Inverse-Gamma predictive model and, for VIX only, Gaussianises standardized returns before applying the model.Its run-length recursion uses hazard H = 1/250, truncates at Rmax = 1500, and prunes hypotheses below 10^-4.Source: §E.1 · pp. 29–30

E.2 Sticky HDP-SLDS

  • The sticky HDP-SLDS benchmark uses a Markov-switching stochastic-volatility model with a sticky HDP-HMM mode sequence and truncation level L = 15.Its observation noise is approximated by a 10-component Gaussian mixture fitted to the log χ2(1) density.Source: §E.2 · p. 30
  • The sticky HDP-SLDS uses blocked Gibbs updates for latent states, noise components, parameters, regime indicators, transition rows, and global weights.The simulation runs 1,500 iterations with 500 burn-in; VIX runs 3,000 with 1,000 burn-in.Source: §E.2 · p. 30

E.3 Martinez–Mena: Pitman–Yor–OU change-point detection

  • Martínez and Mena [2014] use the same Pitman–Yor composition prior as TPGPMM but an OU within-block likelihood and split–merge–shuffle MCMC.The method integrates regime means and precisions under a Normal–Gamma prior; Model A estimates autocorrelation, whereas Model B fixes ϕ = 0, and both fix σ = 0.9.Source: §E.3 · pp. 30–31
  • Change-points are declared when marginal boundary probability exceeds 0.5, with the same 5-step cooldown used by BOCPD and HDP-SLDS.For each retained draw, regime predictions use Student-t distributions; Model A predicts one step ahead because it has no observation noise.Source: §E.3 · p. 31

E.4 Gaussian-process benchmarks

  • GP benchmarks comprise stationary RBF or Matérn-3/2 models fit by marginal likelihood, laGP local predictions, and TPGPMM forecasts averaged over 500 posterior draws.Each TPGPMM forecast uses at most the last 150 observations of the final regime.Source: §E.4 · p. 31

F.1 Posterior-predictive metrics

  • Predictive evaluation uses RMSE, NLPD, and CRPS; scale-normalised scores compare against constant-Gaussian or constant-mean baselines.Lower NLPD and CRPS are better; CRPS remains finite under heavy-tailed errors, and negative ΔNLPD indicates improvement over the marginal-Gaussian baseline.Source: §F.1 · p. 32

F.2 Change-point detection metrics

  • Change-point evaluation matches detections to true boundaries within 5 trading days and reports inferred regime count, precision, recall, F1, and CP_MAE.F1 balances precision and recall; CP_MAE is in trading days and is zero when both boundary sets are empty.Source: §F.2 · pp. 32–33

F.3 TPGPMM-specific summaries

  • Simulation summaries report empirical posterior quantiles for Pitman–Yor parameters, with reported posterior means averaged across replicate seeds.The same seed-averaging convention applies to θ and σ_n when reported.Source: §F.3 · p. 33

G Simulation-study configuration

  • Simulation data use 10 GP blocks with differing means and kernels, tested under equal or power-law lengths; 50 replicates per scenario are fitted for comparison.The fitted TPGPMM lacks block-specific means, and the comparison uses a 5-day matching tolerance with adjacent detections within 4 days merged.Source: G Simulation-study configuration · pp. 33–34

H Sampler diagnostics

The sampler’s chains mix consistently across starting partitions for the main model and most alternatives, while the estimated-noise RBF model mixes less reliably. Verification also checks prior recovery, likelihood calculations, and agreement with an exact sampler.

H Sampler diagnostics

  • TPGPMM’s maximum bR is 1.031 on VIX and 1.011 on Treasury yields, with ESS for K of 143 and 387; the VIX-chain traces overlap across panels.The estimated-noise RBF alternative is the exception: bR is 1.27 on VIX and 1.14 on Treasury yields, with ESS_K = 12 on VIX, consistent with weak separability of amplitude and noise.Source: H Sampler diagnostics · pp. 34–35, Figure 7 · p. 35
  • A basic split–merge sampler had ESS_K = 5 and split-bR = 1.51 on a 20,000-iteration VIX chain; the improved sampler adds exact boundary Gibbs draws, informed split proposals, and dimension-matched splits.The comparison attributes the difference to those three sampler features.Source: H Sampler diagnostics · p. 35

H.1 Verification of the sampler

  • With the likelihood switched off, four chains reproduce the prior closely: the total-variation distance for K is 0.014, and the Kolmogorov–Smirnov distances for σ and θ are 0.019 and 0.002.Toeplitz-prefix block likelihoods match direct multivariate-normal densities to relative error 10^-13 across kernels, with and without regime means.Source: §H.1 · p. 35
  • With the likelihood switched on, the MCMC chains agree with an exact sampler on the posterior of the regime count, boundary probabilities, and Pitman–Yor parameters.Source: §H.1 · p. 35

I Validation against an exact sampler

An exact grid sampler provides a reference for checking whether TPGPMM’s chains recover the posterior under the main specification. The comparison finds close agreement in regime counts, boundary probabilities, and Pitman–Yor parameters, with remaining differences attributed to Monte Carlo error and hyperprior discretisation.

I Validation against an exact sampler

  • The exact sampler discretises the log-normal hyperpriors over kernel parameters and sums block hyperparameters out block by block; refinement approaches the continuous-model posterior.A shared Toeplitz covariance matrix and one factorisation and triangular solve per grid point produce likelihoods for all blocks; forward recursion and backward sampling draw partitions.Source: I Validation against an exact sampler · p. 36
  • The total-variation distances between exact-grid and chain posteriors of K are 0.049 for VIX and 0.024 for the Treasury yield.The exact posterior means of K are 13.69 and 7.00 on the finest grid, versus 13.44 and 6.95 for the chains, respectively.Source: I Validation against an exact sampler · p. 36
  • Mean absolute differences in boundary probabilities are 0.007 for VIX and 0.002 for the Treasury yield; maxima are 0.14 and 0.04.Boundary probability refers to a boundary within ±10 trading days of each date.Source: I Validation against an exact sampler · p. 37, Figure 8 · p. 38
  • Exact-grid posterior means of σ and θ closely match chain means for both series: 0.082 and 3.58 versus 0.090 and 3.56 for VIX, and 0.367 and 5.16 versus 0.375 and 5.26 for Treasury yields.Source: I Validation against an exact sampler · p. 37
  • The coarsest 210-point grid overstates VIX’s regime count at 14.43; finer grids are closer, while all three grids agree with chains for Treasury yields.The finer-grid differences are about as large as their differences from the chains, indicating a remaining gap on the order of discretisation error.Source: I Validation against an exact sampler · p. 37
  • The reference check uses a grid over kernel hyperparameters; estimating observation noise would add a third grid dimension and increase block-likelihood matrix cost linearly with grid size.The exact-grid method uses independent draws and its only approximation is the grid.Source: I Validation against an exact sampler · pp. 35, 37
  • The chains converge to the exact posterior up to Monte Carlo error and hyperprior discretisation, with close agreement in regime counts and boundary probabilities.The comparison is limited to the main specification.Source: I Validation against an exact sampler · pp. 36–37, Table 12 · p. 37

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