{"id":"f785e5fd-8c7d-40e5-9d76-3255fa77c089","arxiv_id":"2501.15155","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A unified framework for time-changed Markov processes shows how to accelerate MCMC convergence while preserving the target distribution, unifying several known algorithms.","lead":"This paper shows that many Markov chain Monte Carlo algorithms can be viewed as time-changed versions of simpler processes, where time speeds up in some regions and slows down in others. The authors prove general conditions under which these time-changed processes target the correct distribution and converge faster, which unifies several known samplers.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4's ergodicity and FCLT conclusions depend on V∈A (Definition 4.2) and aperiodicity (Assumption 4.3), which are assumed but not proved for the examples; Theorem 2.3 itself is unaffected.","rationale":"I agree with the reader's weakest assumption. The concern is load-bearing because the paper's advertised contribution is not merely invariance/LLN (which is solid) but qualitative convergence rates. The rate results all route through Definition 4.2 and Assumption 4.3. The paper explicitly acknowledges both are assumptions/conjectures, so flagging them is faithful to the text. I do not see a more fundamental flaw: Theorem 2.3's proof by change of variables is valid under Assumption 2.2, the PDMP/diffusion characterisations are consistent, and the numerical sections are clearly heuristic. The appropriate verdict is therefore conditional acceptance, pending verification of the A-membership and aperiodicity for the examples. This does not change the reader's verdict.","tokens_in":27545,"tokens_out":14723,"duration_ms":143031,"concrete_test":"For the time-changed ZZP in Proposition 3.4, take V as defined in Eq. (22). Compute L V using the time-changed generator from Theorem 3.1/Prop 3.4 and compute s \\tilde L V using the standard ZZP generator. Check that they are equal as functions on Rd×{±1}d and that V lies in the domain of both generators under Assumption 4.8. If equality holds, Prop 4.9's geometric ergodicity step survives; if not, the V∈A assumption fails in the running example. The same substitution can be repeated for \\bar V=V/(1+V) to test Theorem 4.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central invariance/LLN result (Theorem 2.3) is well supported. The strong claim about convergence rates is carried by Theorems 4.4-4.6 and their applications. Every one of those results requires the Lyapunov function V of the base process to lie in A, i.e. L V = s \\tilde L V. The paper states V∈A as an assumption (Assumption 4.8 for the ZZP) and asserts aperiodicity of X (Assumption 4.3), which is explicitly left as a conjecture for the general PDMP examples. Proposition B.1 only establishes A-membership for bounded functions with bounded s \\tilde L f, so it does not cover the unbounded exponential Lyapunov functions used in the ZZP applications. If V∉A, the drift inequality LV≤-sWV+... used in the proof of Theorem 4.4 does not follow, and the geometric/uniform ergodicity and FCLT conclusions are unsupported. The paper's own text flags this: 'we shall then assume that V∈A' and 'We conjecture that Assumption 4.3 holds...'. This is a genuine technical gap, not a disagreement with consensus: the invariance/LLN part is fine, but the advertised rate-control contribution needs the missing membership checks.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a general construction for MCMC samplers based on random time changes of a Markov process. Given a base process Y targeting ~μ ∝ s μ with a user-chosen 'speed function' s, the time-changed process X_t = Y_{r(t)} with clock r(t) = ∫_0^t s(X_u) du (Eq. (1)) is shown in Theorem 2.3 to have μ as its unique stationary distribution and to satisfy a strong LLN for every f ∈ L^1(μ), provided Y satisfies an LLN with respect to ~μ. The construction is applied to PDMPs (Zig-Zag, Bouncy Particle Sampler, Randomized HMC), to overdamped and underdamped Langevin diffusions, and to discrete-state jump processes; Proposition 3.4 recovers the speed-up ZZP of Vasdekis and Roberts [2023] and Example 3.8 recovers the time-changed Langevin dynamics of Roberts and Stramer [2002]. Section 4 develops criteria for geometric and uniform ergodicity (Theorems 4.4, 4.5, 4.13) and a functional central limit theorem with an importance-sampling-like variance formula (Theorem 4.6, Eqs. (14)–(15)), all under drift, petite-set, domain-membership (V ∈ A), and aperiodicity conditions. Section 5 relates time changes to space transformations, and Section 6 reports toy simulations for a multimodal Gaussian mixture and a heavy-tailed rare-event problem.","tokens_in":27864,"tokens_out":12851,"duration_ms":104507,"significance":"The construction is conceptually appealing and unifies several known algorithms, connecting time changes with pathwise importance sampling and with Jacobian-based space transformations; the relationship to prior work is drawn carefully, and credit is explicitly given where algorithms are re-discovered. Theorem 2.3 is the clean centerpiece: its assumptions are nearly minimal, and the proof is a short change-of-variables argument that also justifies the reweighted estimators of Section 2.2. The paper is unusually explicit about what is assumed versus proved, flags the V ∈ A condition and the aperiodicity conjecture, and ships reproducible code. If the Section 4 hypotheses are verified for the examples, the rate-control message would be a genuine contribution to continuous-time MCMC; the open questions identified below concern exactly that verification.","major_comments":[{"comment":"The ergodicity statements for the running example hinge on the membership V ∈ A of Definition 4.2, yet this membership is never established; the paper's own Section 4.1 states 'we shall then assume that V ∈ A.' Proposition B.1 only puts bounded functions f with bounded s L~ f into A, whereas the ZZP Lyapunov function in Eq. (22) is unbounded (exponential in α U~ plus a sum of φ terms), and the drift condition (12) forces L~ V to be unbounded as well. The proof of Theorem 4.4 (Appendix D.1) begins with the identity L V = s L~ V, which for the ZZP application is exactly the content of Assumption 4.8 that Proposition 4.9 assumes without verification. Since the advertised rate-control contribution rests on Theorems 4.4–4.6, the paper should either prove V ∈ A under the conditions of Assumption 4.8 or give checkable sufficient conditions, for instance through the PDMP martingale problem, and verify them for the time-changed ZZP.","section":"§4.1 (Definition 4.2), §4.3 (Assumption 4.8), Appendix D.1"},{"comment":"The FCLT statement and the variance identity (15) depend on two conditions that Proposition 4.11 does not justify. First, the proof in Appendix E uses 'ĝ ∈ A' to identify the solution of the Poisson equation for the time-changed generator with that of the base generator; for the time-changed ZZP one must check that the solution of g = −L ĝ lies in A, which is not done. Second, the global bound s(x)W(x)V(x) ≥ 1 for all x ∈ E is asserted in Theorem 4.6 but not verified in the ZZP setting with s(x) = exp(βU(x)) and the exponential V of Eq. (22). Until these hypotheses are verified, Proposition 4.11 is conditional on the same unresolved domain-membership issue as Proposition 4.9.","section":"§4.2 (Theorem 4.6), §4.3 (Proposition 4.11), Appendix E"},{"comment":"Assumption 4.3 (aperiodicity of the time-changed process) is assumed throughout Section 4, and the paper itself calls it a conjecture in the general case ('We conjecture that Assumption 4.3 holds under aperiodicity of Y and some regularity conditions'), with a proof cited only for the time-changed ZZP. Theorems 4.5 and 4.6 are advertised as applicable to the other PDMP examples of Section 3.1.2 (BPS, RHMC), but no aperiodicity argument is supplied for those processes, so those applications do not yet satisfy the stated hypotheses. The paper should either restrict the general claims, prove Assumption 4.3 under the stated regularity conditions, or mark explicitly which examples satisfy all hypotheses.","section":"Assumption 4.3, §3.1.2, Theorems 4.4–4.6"}],"minor_comments":[{"comment":"In Eq. (11) the total-variation distance is written with the base process's stationary distribution ~μ; since X targets μ, this should read μ, and the same slip appears in the surrounding text.","section":"§4.1, Eq. (11)"},{"comment":"There are a few typos: 'streightforward' and 'Howerver' in Section 2.2, and 'FLCT' in Proposition 4.14, which should read 'FCLT'.","section":"§2.2, Proposition 4.14"},{"comment":"Roberts and Stramer [2002a] and [2002b] appear to be the same article listed twice in the bibliography; the citations in the Introduction and in Example 3.8 should point to a single entry.","section":"References"},{"comment":"In the numerical experiments the speed exponent a is tuned by hand (a = 0.9 in Figure 1 and in Figure 4b, a = 0.3 in Figure 5b), and the illustrative traces are shown without error bars or a sensitivity discussion; a brief explanation of the choice of a and of the stability of the conclusions in a would make the empirical support more convincing, although Figure 5 does quantify MSE over 5000 runs.","section":"§6, Figures 1, 2, 4, 5"},{"comment":"The notation s_C is used both for the supremum of s over the set D and in the definition of t1 = t0 / s_C, which collides with the lower bound s of Assumption 2.1; the two objects should be given distinct names.","section":"Appendix B.3, Lemma B.3"},{"comment":"The abstract contains two consecutive sentences announcing numerical simulations ('Throughout the paper we give various visualisations...' and 'Finally, we provide numerical simulations...'); one of the two should be removed to avoid redundancy.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The main gap (V ∈ A for the ZZP Lyapunov function) is not a hidden error: the paper states it as an assumption and tells the reader where it will be assumed. Nevertheless, it is load-bearing for the paper's rate-control claims, so closing it requires genuine mathematical work rather than editorial patching. I would be comfortable with acceptance once the authors either verify A-membership under Assumption 4.8 or reframe the theorems as conditional on a checkable condition, and once aperiodicity for the BPS/RHMC applications is either proved or explicitly removed from the claims. The paper fits the journal's scope, the novelty disclosure is fair, and the central invariance/LLN result is sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing you should know: this paper is a genuine unification, not just a repackaging. Theorem 2.3, which says that a time-changed process targets the reweighted measure whenever the base process has an LLN under the tilted measure, is clean and useful. It subsumes the Speed-Up Zig-Zag, time-changed Langevin diffusions, and importance sampling on paths, and it makes the connection to space transformations explicit. The paper is well written, gives a working algorithm for jump processes, and ships code. That part I buy.\n\nThe softer spots are where the advertised rate-control results live. Section 4's geometric/uniform ergodicity and FCLT theorems all require V ∈ A (Definition 4.2) and aperiodicity (Assumption 4.3). The paper is honest about these being assumptions rather than theorems: it says \"we shall then assume that V ∈ A\" and \"we conjecture that Assumption 4.3 holds.\" But the consequence is that Proposition 4.9 and Proposition 4.11, the actual payoff for the Zig-Zag example, inherit those unverified membership conditions. Proposition B.1 only establishes A-membership for bounded functions with bounded s L̃ f, so it does not cover the unbounded exponential Lyapunov functions used in the applications. That's a real gap, though not a fatal one: the invariance and LLN result does not depend on it, and the ergodicity/FCLT conclusions would become theorems if the membership could be checked for these examples.\n\nThe numerical sections are illustrative rather than exhaustive. There are no error bars on the trace plots, and the speed exponent a in s(x)=exp(aU(x)) is tuned without a clear adaptive or data-driven rule. Given the theory gap, I'd treat the numerics as proof-of-concept, not as strong empirical evidence.\n\nOverall: this paper deserves a serious referee. It advances the field by giving a common language to algorithms that previously seemed separate, and the central theorem is solid. The referee should push for either verifying V ∈ A and aperiodicity for the Zig-Zag and BPS/RHMC examples, or reformulating the theorems so those conditions become explicit hypotheses whose satisfaction is a concrete open problem. I'd cite it for the framework and the invariance theorem; I'd be more careful about citing the ergodicity claims until the assumptions are sorted out.","headline":"Solid unifying framework for time-changed MCMC; the core invariance/LLN theorem is clean, but the stronger ergodicity claims rest on unverified technical assumptions.","tokens_in":28327,"tokens_out":1553,"would_cite":true,"duration_ms":16302,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J25","60J60","65C05","65C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Random time-changes turn Markov processes with a law of large numbers into samplers for any reweighted target, with the speed function controlling convergence.","keywords":["time-changed Markov processes","Markov chain Monte Carlo","piecewise deterministic Markov processes","Zig-Zag process","geometric ergodicity","functional central limit theorem","importance sampling","Langevin diffusion"],"falsifier":"Simulate the time-changed Zig-Zag process on the two-component Gaussian mixture of Section 6.1 with speed $s(x)=\\mu(x)^{-0.9}$ and measure the long-run fraction of time spent in each component over increasing horizons $T$. Theorem 2.3 predicts the fractions converge to the mixture weights; if the discrepancy does not shrink as $T$ grows, the LLN/invariance claim fails. A sharper test of the ergodicity results: compute $LV$ for the explicit $V$ in equation (22) for a non-quadratic target; the proof of Proposition 4.9 needs $LV=s\\tilde L V$ outside a compact set, so any state where this identity fails would refute the claimed geometric ergodicity for that speed.","tokens_in":27379,"feed_emoji":"⏱️","tokens_out":10688,"duration_ms":93209,"temperature":0.7,"pith_summary":"This paper tries to establish that sampling from a target distribution $\\mu$ can be reduced to choosing a speed function $s$ for a well-understood Markov process $Y$. The time-changed process $X_t=Y_{r(t)}$, with $r(t)=\\int_0^t s(X_u)\\,du$, follows the same paths as $Y$ but runs faster where $s$ is large. Theorem 2.3 shows that if $Y$ satisfies a strong law of large numbers for the reweighted distribution $\\tilde\\mu(dx)\\propto s(x)\\mu(dx)$, then $X$ has $\\mu$ as its unique stationary distribution and satisfies a strong law of large numbers for every $f\\in L^1(\\mu)$. The authors use this to construct time-changed versions of Langevin diffusions and piecewise deterministic Markov processes, recovering the Speed-Up Zig-Zag process and producing new samplers, and they give conditions under which the time-changed process is geometrically or uniformly ergodic and obeys a functional central limit theorem. A reader should care because it turns a single base process into a tunable sampler for a family of targets, with the speed function as the dial controlling convergence.","feed_headline":"A speed function makes any LLN sampler target a reweighted law","feed_subtitle":"Changing the clock by a state-dependent speed preserves laws of large numbers and can accelerate convergence into hard regions.","key_machinery":"The load-bearing object is the random time-change $r(t)=\\int_0^t s(X_u)\\,du$ and its inverse relation $Y_t=X_{r^{-1}(t)}$. The paper's key identity is the generator scaling $L=s\\tilde L$ on a set $A$ of functions in both generator domains; this is what lets ergodicity and CLT results flow from $Y$ to $X$. The invariance proof itself rests on the change of variables $\\frac1T\\int_0^T f(X_t)\\,dt=\\frac{r(T)}T\\frac1{r(T)}\\int_0^{r(T)} \\frac{f(Y_u)}{s(Y_u)}\\,du$, where $r(T)/T$ converges almost surely to $\\mu(s)$ by the base LLN. For the drift arguments, a Lyapunov function $V$ with $\\tilde L V\\le -WV+\\gamma 1_C$ is turned into $LV\\le -sWV+s\\gamma 1_C$, so the speed function appears directly in the exponential rate.","core_discovery":"The central discovery is that the time-change is, in essence, a pathwise importance reweighting: speeding up time where $s$ is large is the same as reweighting the base process's path by $1/s$. Because of this, the time-changed process inherits the base process's law of large numbers provided the base process targets the $s$-biased distribution, and its ergodic averages converge to $\\mu$. The authors further show that under drift conditions on $Y$ the generator identity $L=s\\tilde L$ lets the speed function repair slow convergence: requiring $s(x)\\ge\\zeta/W(x)$ outside a compact set converts geometric ergodicity, and stronger lower bounds on $s$ give uniform ergodicity even for base processes that only converge polynomially. They also obtain a functional central limit theorem whose asymptotic variance is, up to the constant $\\mu(s)$, the base process's variance for the observable $g/s$.","pith_inferences":["Going beyond the paper: the self-normalised importance-sampling analogy suggests that the optimal speed for estimating a fixed observable $g$ should be $s(x)\\propto|g(x)-\\mu(g)|$, an analogue of the optimal IS proposal that the paper mentions but does not prove.","Going beyond the paper: the Section 5 observation that Jacobian determinants act as speed functions points to a systematic source of new speed functions—any diffeomorphism from a compact domain to $\\mathbb{R}^d$, such as the stereographic projection, yields a speed function and a corresponding time-changed sampler.","Going beyond the paper: because the speed function can be any positive lower-bounded function, one could tune it adaptively during a run, alternating exploration of modes with sampling of tails; the paper lists this as future work, not a demonstrated method."],"forward_implications":["Every continuous-time MCMC algorithm whose base process satisfies an LLN can be re-purposed as a sampler for any absolutely continuous reweighting of its target by choosing a lower-bounded speed function.","Known algorithms such as the Speed-Up Zig-Zag are time-changes of a standard Zig-Zag, so their LLN and ergodicity properties transfer from the base process without ad-hoc proofs.","For a PDMP, the time-changed process is again a PDMP with vector field and rate multiplied by $s$ and unchanged jump kernel; this yields new $\\mu$-stationary BPS and RHMC variants.","The FCLT relation $\\gamma_g^2=\\mu(s)\\tilde\\gamma_{\\tilde g}^2$ makes variance comparisons between samplers a question of choosing $s$, mirroring self-normalised importance sampling.","A jump-process implementation (Algorithm 1) with rate $s$ and a kernel leaving $\\tilde\\mu$ invariant can be simulated with only exponential clocks and gives asymptotically unbiased estimates."],"supporting_citations":[{"why":"First considered a time-changed overdamped Langevin diffusion with speed $\\mu^{-\\alpha}$; the paper generalises this specific construction.","marker":"Roberts and Stramer [2002a]"},{"why":"Introduced the Speed-Up Zig-Zag that the paper re-derives as a time-changed ZZP, and supplies aperiodicity and ergodicity facts used in the applications.","marker":"Vasdekis and Roberts [2023]"},{"why":"Defines the Zig-Zag process used as the running example and source of the base-process LLN.","marker":"Bierkens et al. [2019a]"},{"why":"Provides the LLN and drift/Lyapunov conditions for the Zig-Zag process on which Proposition 4.9 rests.","marker":"Bierkens et al. [2019b]"},{"why":"Supplies the drift and petite-set criteria that Theorems 4.4 and 4.5 invoke for geometric and uniform ergodicity.","marker":"Down et al. [1995]"},{"why":"Gives the Poisson-equation and CLT machinery behind Theorem 4.6 and the variance formula (15).","marker":"Glynn and Meyn [1996]"},{"why":"Connects the reweighted-path estimator (6) to optimal importance sampling for overdamped Langevin dynamics.","marker":"Chak et al. [2023]"},{"why":"Provides the space-transformation framework compared with time-changes in Section 5.","marker":"Johnson and Geyer [2012]"}],"fun_headline_variants":["Time-changed Markov processes speed MCMC convergence","A speed function reweights paths to sharpen MCMC mixing","Warp time to make MCMC reach rare states faster","Faster MCMC: adjust the clock to reweight paths","State-dependent speeds accelerate MCMC into hard modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the drift certificate for the base process—a function $V$ used to prove its ergodicity—lies in the class where the time-changed generator is exactly $s$ times the base generator, and that the time-changed process is aperiodic; the paper assumes both and notes that aperiodicity is only conjectured in general.","fun_headline_variants_meta":{"raw":{"variants":["Time-changed Markov processes speed MCMC convergence","A speed function reweights paths to sharpen MCMC mixing","Warp time to make MCMC reach rare states faster","Faster MCMC: adjust the clock to reweight paths","State-dependent speeds accelerate MCMC into hard modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1481,"prompt_tokens":928,"completion_tokens":553,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":475}},"tokens_in":544,"tokens_out":553,"duration_ms":5655,"temperature":1.0,"reasoning_tokens":475,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:33:58.374868+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the time-changed Zig-Zag process on the two-component Gaussian mixture of Section 6.1 with speed $s(x)=\\mu(x)^{-0.9}$ and measure the long-run fraction of time spent in each component over increasing horizons $T$. Theorem 2.3 predicts the fractions converge to the mixture weights; if the discrepancy does not shrink as $T$ grows, the LLN/invariance claim fails. A sharper test of the ergodicity results: compute $LV$ for the explicit $V$ in equation (22) for a non-quadratic target; the proof of Proposition 4.9 needs $LV=s\\tilde L V$ outside a compact set, so any state where this identity fails would refute the claimed geometric ergodicity for that speed.","supporting_citations":[],"review_version":1}