Broadly Applicable Approximate MCMC for Switching Stochastic Differential Equations Using Uniformization and Time-Conditioned Factorized Neural Likelihood Estimation

Shion Hosoda; Michiaki Hamada

stat.MLOct 7, 2026

Broadly Applicable Approximate MCMC for Switching Stochastic Differential Equations Using Uniformization and Time-Conditioned Factorized Neural Likelihood Estimation

Authors: Shion Hosoda, Michiaki Hamada

Organizations: Waseda University · National Institute of Advanced Industrial Science and Technology · Nippon Medical School

Abstract

Switching stochastic differential equations (SSDEs) describe continuous-time dynamics whose parameters switch according to a latent regime process that follows a continuous-time Markov chain (CTMC). By allowing dynamics to change between regimes, SSDEs represent heterogeneous system behavior and have been applied across diverse fields. However, Bayesian inference for SSDEs remains difficult, and existing SSDE inference methods have limited applicability, with restrictions such as noise-free observations, univariate states, linear drift, or state-independent diffusion. In this study, we propose an approximate Markov chain Monte Carlo sampler for SSDEs using uniformization and factorized neural likelihood estimation (FNLE), a simulation-based inference method. Uniformization provides an exact representation of the CTMC but requires SDE transition densities over arbitrary time intervals. We approximate these densities by training a time-conditioned FNLE model. The resulting sampler is broadly applicable to SSDEs without requiring analytically tractable transition densities. In synthetic-data experiments, our method recovered regime paths and parameters for three SSDE models for which previous methods have limited applicability. We also applied our method to a real dataset and detected a regime transition.

Figures & tables

Figure 1: Illustration of a simulated Lotka–Volterra SSDE. The upper panel shows the two components of the continuous state and their noisy observations. Background shading indicates the active regime, and the lower panel shows the regime path.

Method	Observation error	Nonlinear drift	State-dependent diffusion	Intractable transition density	Multivariate states
Blackwell et al. [3]	$\triangle$	$\triangle$	$\triangle$	$\bm{\times}$	$\bm{\checkmark}$
Hibbah et al. [13]	$\triangle$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\triangle$
Köhs et al. [19]	$\bm{\checkmark}$	$\triangle$	$\bm{\times}$	$\triangle$	$\bm{\checkmark}$
Köhs et al. [20]	$\bm{\checkmark}$	$\bm{\times}$	$\bm{\times}$	$\bm{\times}$	$\bm{\checkmark}$
Stumpf-Fétizon et al. [32]	$\triangle$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\triangle$
Proposed method	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$

Table 1: Applicability of SSDE Inference Methods

Figure 2: Comparison of inference results for the OU experiment using an exact MCMC sampler and two approximate MCMC samplers with FNLE trained on exact transitions and Euler–Maruyama simulations, respectively.

Model	Observation error	Nonlinear drift	State-dependent diffusion	Intractable transition density	Multivariate states
Ornstein–Uhlenbeck	$\bm{\checkmark}$	$\bm{\times}$	$\bm{\times}$	$\bm{\times}$	$\bm{\times}$
Lotka–Volterra	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$
Gene-expression CLE	$\bm{\checkmark}$	$\bm{\times}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$
Susceptible–infected–recovered epidemic model	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$	$\bm{\checkmark}$

Table 2: Characteristics of the SSDE Models Used to Generate the Synthetic Datasets

Figure 3: Inference results for synthetic data from three SSDEs with regime-specific Lotka–Volterra, gene-expression chemical Langevin, and susceptible–infected–recovered epidemic models (left to right).

Figure 4: Real-data inference results for the Didinium – Paramecium time series using a Lotka–Volterra SSDE.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Experiment	$n_{\max}$	$n_{\max}h$
Synthetic-data OU (approx. MCMC with Euler–Maruyama FNLE)	100	1.00
Synthetic-data LV	100	1.00
Synthetic-data gene-expression CLE	25	0.25
Synthetic-data susceptible–infected–recovered	20	0.20
Real-data LV	98	0.98

Appendix

Table 3: Experiment-Specific FNLE Training Settings.

Experiment	Component	Prior
Synthetic-data OU	scalar state	$\tau^{2}\sim\mathrm{InvGamma}(10^{-3},10^{-3})$
Synthetic-data LV	predator	$\tau_{1}^{2}\sim\mathrm{InvGamma}(10^{-3},10^{-3})$
Synthetic-data LV	prey	$\tau_{2}^{2}\sim\mathrm{InvGamma}(10^{-3},10^{-3})$
Synthetic-data gene-expression CLE	mRNA	$\tau_{1}^{2}\sim\mathrm{InvGamma}(10^{-3},10^{-3})$
Synthetic-data gene-expression CLE	protein fluorescence	$\tau_{2}^{2}\sim\mathrm{InvGamma}(10^{-3},10^{-3})$
Synthetic-data susceptible–infected–recovered	susceptible	$\tau_{1}^{2}\sim\mathrm{InvGamma}(10^{-3},10^{-3})$

Appendix

Table 4: Priors for the Observation Variances

Experiment	State path $y_{1:L}$	Parameters $\Theta$
Synthetic-data OU (exact MCMC)	$2^{-8}$	$2^{-5}$
Synthetic-data OU (approx. MCMC with exact FNLE)	$2^{-8}$	$2^{-5}$
Synthetic-data OU (approx. MCMC with Euler–Maruyama FNLE)	$2^{-8}$	$2^{-5}$
Synthetic-data LV	$2^{-9}$	$2^{-8}$
Synthetic-data gene-expression CLE	$2^{-9}$	$2^{-8}$
Synthetic-data susceptible–infected–recovered	$2^{-9}$	$2^{-8}$

Appendix

Table 5: Fixed NUTS Step Sizes by Experiment and Sampler

Experiment	Chains	Sweeps	Burn-in	Retained
Synthetic-data OU (exact MCMC)	4	10,000	5,000	5,000
Synthetic-data OU (approx. MCMC with exact FNLE)	4	10,000	5,000	5,000
Synthetic-data OU (approx. MCMC with Euler–Maruyama FNLE)	4	10,000	5,000	5,000
Synthetic-data LV	4	10,000	5,000	5,000
Synthetic-data gene-expression CLE	4	10,000	5,000	5,000
Synthetic-data susceptible–infected–recovered	4	10,000	5,000	5,000

Appendix

Table 6: MCMC Chain Counts and Lengths Used for Posterior Summaries. Sweep, burn-in, and retained-draw counts are per chain.

Model	Parameter	Regime 1	Regime 2	Structure
OU	$\kappa$	0.15	0.15	shared
	$\mu$	-1.2	1.2	regime-specific
	$\sigma$	0.05	0.05	shared
LV	$\alpha$	1.0	0.5	regime-specific
	$\beta$	1.0	0.5	regime-specific
	$\gamma$	1.0	0.2	regime-specific

Appendix

Table 7: Data-Generating SDE Parameters for the Synthetic Experiments. Regime-specific values are listed for regimes 1 and 2.

Model	Parameter	Structure	FNLE range	Prior
OU	$\kappa$	shared	$[0.05,0.2]$	$\operatorname{LogNormal}(0,1)$
	$\mu$	regime-specific	$[-2.0,2.0]$	$\operatorname{Normal}(0,1)$
	$\sigma$	shared	$[0.01,0.1]$	$\operatorname{LogNormal}(0,1)$
LV	$\alpha$	regime-specific	$[0.01,2]$	$\operatorname{LogNormal}(0,1)$
	$\beta$	regime-specific	$[0.1,3]$	$\operatorname{LogNormal}(0,1)$
	$\gamma$	regime-specific	$[0.01,2]$	$\operatorname{LogNormal}(0,1)$

Appendix

Table 8: SDE Parameter Inference Settings for the Synthetic Experiments. Parameter ranges and priors are on the physical scale.

Parameter	Structure	FNLE range	Prior
$\alpha$	regime-specific	$[0.1,4]$	$\operatorname{LogNormal}(0,1)$
$\beta$	regime-specific	$[0.01,6]$	$\operatorname{LogNormal}(0,1)$
$\gamma$	regime-specific	$[0.1,8]$	$\operatorname{LogNormal}(0,1)$
$\delta$	regime-specific	$[0.01,5]$	$\operatorname{LogNormal}(0,1)$
$\sigma_{1}$	shared	$[0.01,0.8]$	$\operatorname{LogNormal}(-1,1)$
$\sigma_{2}$	shared	$[0.01,2]$	$\operatorname{LogNormal}(-1,1)$

Appendix

Table 9: SDE Parameter Settings for the Real-Data LV Analysis. Parameter ranges and priors are on the physical scale.

Figure 5: Multi-chain diagnostics for the real-data Lotka–Volterra analysis. Left panels show chain-specific marginal posterior density estimates, and right panels show traces over all 10,000 sweeps. Colors distinguish the four chains; solid and dotted lines distinguish regimes 1 and 2 for regime-specific parameters. Vertical dashed lines mark the burn-in cutoff at 5,000 sweeps. The observation-noise panels show standard deviations $\tau_{1},\tau_{2}$ , not variances.

OU sampler	Parameter	$\hat{R}$	Bulk ESS	Tail ESS
Exact MCMC	$\kappa$	1.015	198.4	1,004.2
	$\mu_{1}$	1.004	1,187.8	6,199.0
	$\mu_{2}$	1.005	1,478.3	5,050.6
	$\sigma$	1.004	1,519.8	2,234.7
	$q_{12}$	1.000	16,311.0	19,035.3
	$q_{21}$	1.000	16,972.1	18,373.4

Appendix

Table 10: Per-parameter MCMC diagnostics for the three OU samplers on synthetic data.

Parameter	$\hat{R}$	Bulk ESS	Tail ESS
$\alpha_{1}$	1.006	421.2	1,090.5
$\alpha_{2}$	1.007	406.2	777.6
$\beta_{1}$	1.006	395.9	973.0
$\beta_{2}$	1.006	385.9	771.1
$\gamma_{1}$	1.028	217.1	893.4
$\gamma_{2}$	1.038	143.4	301.5

Appendix

Table 11: Per-parameter MCMC diagnostics for the synthetic Lotka–Volterra experiment.

Parameter	$\hat{R}$	Bulk ESS	Tail ESS
$\rho_{1}$	1.003	805.9	1,595.1
$\rho_{2}$	1.001	1,274.5	3,142.3
$\beta$	1.011	211.0	333.4
$\gamma$	1.007	334.2	691.6
$\delta$	1.007	335.3	681.7
$c$	1.018	201.7	358.9

Appendix

Table 12: Per-parameter MCMC diagnostics for the gene-expression CLE experiment.

Parameter	$\hat{R}$	Bulk ESS	Tail ESS
$\beta_{1}$	1.019	481.9	1,013.8
$\beta_{2}$	1.054	61.6	338.2
$\gamma_{1}$	1.065	66.2	485.9
$\gamma_{2}$	1.003	645.6	1,278.9
$q_{12}$	1.000	15,315.1	18,870.5
$q_{21}$	1.001	10,209.2	14,825.5

Appendix

Table 13: Per-parameter MCMC diagnostics for the susceptible–infected–recovered experiment.

Parameter	$\hat{R}$	Bulk ESS	Tail ESS
$\alpha_{1}$	1.004	442.4	987.0
$\alpha_{2}$	1.005	929.8	909.5
$\beta_{1}$	1.003	466.3	1,065.3
$\beta_{2}$	1.005	871.0	888.9
$\gamma_{1}$	1.011	298.2	732.9
$\gamma_{2}$	1.007	1,160.2	2,167.1

Appendix

Table 14: Per-parameter MCMC diagnostics for the real-data Lotka–Volterra analysis.

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Stochastic differential equations (SDEs) provide a flexible framework for modeling temporal dynamics in partially observed systems. A central task is to calibrate such models from data, which requires inferring latent trajectories and parameters from sparse, noisy observations. Classical smoothing methods for this problem are often limited by path degeneracy and poor scalability. In this work, we developed a novel method based on characterization of the posterior SDE in terms of conditional backward-in-time score defined as the gradient of a function solving a Kolmogorov backward equation with multiplicative updates at observation times. We learn this conditional score using neural networks trained to satisfy both the governing PDE and the observation-induced jump conditions, thereby integrating continuous-time dynamics with discrete Bayesian updates. The resulting score induces a posterior SDE with the same diffusion coefficient but a modified drift, enabling efficient posterior trajectory sampling. We further derive a likelihood-based objective for learning the SDE parameters, yielding an evidence lower bound (ELBO) for joint state smoothing and parameter estimation. This leads to a variational EM-style procedure, where the neural conditional score is optimized to approximate the smoothing distribution, followed by a maximization step over the SDE parameters using samples from the induced posterior. Experiments on nonlinear systems demonstrate accurate and stable inference with a very few observations demonstrating significant improved scalability compared to classical MCMC methods.

Abstract

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