Discrete Markov Bridges: Theory & Applications

• Discrete Markov bridges are stochastic processes conditioned to start and end at specified states using Doob’s h-transform.
• They blend reciprocal invariants with discrete Markov properties to enable precise simulation, rare-event analysis, and robust inference.
• Modern applications leverage these bridges for efficient generative modeling in fields such as molecular design and computational biology.

A discrete Markov bridge is a stochastic process—a Markov chain or discrete-time jump process—conditioned to start and end at specified states (or distributions) over a fixed number of steps. Discrete Markov bridges retain many key properties and construction principles from their continuous-time analogues, such as conditioning via Doob’s h-transform and structure determined by reciprocal invariants, but exhibit unique behaviors and admit specialized modeling frameworks suitable for discrete data, random walks, counting processes, and modern generative modeling. Recent research highlights their centrality in theoretical probability, statistical physics, inference, representation learning, and applied areas such as molecular design and computational biology.

1. Formal Definition and Conditioning Principles

A discrete Markov bridge is a process $(X_n)_{n=0}^N$ on a (finite or countable) state space $\mathcal{X}$, governed by a Markov kernel $p(x, y)$, and conditioned such that $X_0 = x_0$ and $X_N = x_N$ (or, more generally, $X_N$ lies in some set $A$ or has law $\mu$). The law of such a bridge can be constructed via Doob’s $h$-transform, with the $h$-function defined as

$h(n, x) = P^{x}(X_N = x_N \mid X_n = x).$

The transition kernel of the conditioned (bridge) process is

$p^h(x, y) = \frac{p(x, y)\, h(n+1, y)}{h(n, x)},$

ensuring that, under $P^h$, the process is forced, with probability one, to reach the desired terminal state at time $N$ (Çetin et al., 2014).

Conditioning is thus encoded by reweighting transitions with the ratio $h(n+1, y)/h(n, x)$ at each time step, paralleling the continuous SDE theory and ensuring "steering" of the trajectories toward the endpoint.

2. Types of Conditioning and h-Transform Mechanics

Two main types of conditioning appear in the literature:

• Strong conditioning (or "bridge" in the narrow sense): conditioning on a terminal state, $X_N = z$, which usually has nonzero probability in discrete space. The $h$-function specializes to $h(n, x) = P^{x}(X_N = z)$.
• Weak conditioning: conditioning on more general events of positive probability (e.g., $X_N \in A$), where $h(n, x) = P^{x}(X_N \in A \mid X_n = x)$. Both cases employ the $h$-transform for the Markov kernel (Çetin et al., 2014).

In practice, the $h$-function is computed recursively via a backward equation:

$$h(n, x) = \sum_y p(x, y)\, h(n+1, y), \qquad h(N, x) = \mathbf{1}_{\{x_N\}}(x).$$

In applied contexts, this enables simulation of paths conditioned on endpoints, crucial for Monte Carlo inference, rare-event analysis, and stochastic control.

3. Reciprocal Classes and Characterization

Discrete Markov bridges are closely linked to the concept of reciprocal classes: all processes sharing the same family of bridges. For Markov counting processes, the reciprocal class is determined by the reciprocal invariant

$$\Xi_\ell(t, z) = \partial_t \log \ell(t, z) + \ell(t, z+1) - \ell(t, z),$$

where $\ell(t, z)$ is the jump intensity. Two counting processes have the same bridges if and only if their reciprocal invariants coincide (Conforti et al., 2014).

Reciprocal invariants and duality formulas underpin a complete characterization: a process belongs to the reciprocal class of $\ell$ if its dynamics, when viewed through appropriate directional derivatives on path space, match those of the corresponding bridge law.

For Markov chains, a similar role is played by the Doob $h$-transform and, in the context of discrete flux/occupation problems, by the so-called bridge representation of large deviation rate functionals (Renger, 28 Jun 2024).

4. Quantitative and Structural Properties

A range of quantitative properties of discrete Markov bridges have been established:

• Mean path shape: Convexity (lazy bridges) or concavity (hurried bridges) of the bridge mean is determined by the sign of the reciprocal characteristic $\Xi_\ell(t, z)$. If $\Xi_\ell \ge 0$ uniformly, the mean is convex—arrivals cluster near the endpoint; if $\Xi_\ell \le 0$, it is concave—jumps happen early (Conforti, 2015).
• Marginals and jump times: Sharp bounds and tail estimates for the process marginals and inter-jump times can be expressed, in some cases, via binomial distributions or precise large deviation forms (Conforti, 2015, Conforti, 2016).
• Law of large numbers: High bridges (large endpoint differences) converge, after scaling, to deterministic curves determined by $\pi_\ell(t)$, the limit function derived from the reciprocal characteristic (Conforti, 2015).

These structural results provide concrete probabilistic controls and illustrate the dampening or amplification effect of conditioning at the endpoints.

5. Modern Algorithms and Generative Modeling

Discrete Markov bridges have become foundational tools in data-driven, generative models:

• Matrix and Score Learning: In the Discrete Markov Bridge (DMB) approach, the forward process is a time-inhomogeneous CTMC with a learned transition matrix, mapping an initial distribution $\mu$ to a prior, and the backward process is governed by a neural network (score learner) estimating

$$s_\theta(x_t, t)_y \approx \mathbb{E}\left[\frac{p_{t|0}(y|x_0)}{p_{t|0}(x_t|x_0)}\right],$$

used to parametrize the reverse process (Li et al., 26 May 2025). Rigorous L1 conservation properties, accessibility of the target, and convergence in KL-divergence are established theoretically.

• Self-Consistency and Non-Autoregressive Generation: Markov bridges offer latent variable pathways that are more expressive than standard discrete diffusion with fixed matrices. Convergence proofs and efficient parameterizations are possible using upper-triangular structured matrices, reducing space complexity (Li et al., 26 May 2025). Empirically, DMB achieves strong bits-per-character results on text and competitive FID scores on image benchmarks.

These modern generative modeling frameworks systematically exploit the bridge constructions to mitigate error accumulation, enhance sample efficiency, and broaden the design space in discrete data.

6. Applications in Physical, Biological, and Statistical Sciences

Markov bridges have wide-ranging applications:

• Statistical physics: Exact formulas for joint distributions of time-integrated currents and frenesy in cyclic Markov bridges, foundational for fluctuation theorems in nonequilibrium systems; ability to handle absolutely irreversible transitions (Roldán et al., 2019).
• Cellular lineage and stochastic path sampling: Efficient generation of rare transition paths (e.g., between metastable states in single-cell data or molecular potentials) is possible via bridge sampling with time-dependent, state-dependent rate adjustments, enabling detailed analysis of bottlenecks and fate choice (Treut et al., 2023).
• Parameter estimation in jump processes: Time-reversal-based simulation of Markov bridges leads to efficient MCEM and MCMC algorithms for estimating the infinitesimal generators of Markov jump processes, outperforming rejection and uniformization methods especially for long time horizons (Baltazar-Larios et al., 2023).
• Generative and inverse design tasks: Markov bridge frameworks with structure-conditioned priors yield state-of-the-art results in sequence generation for protein design, retrosynthetic planning for chemical reactions, and enable energy-based fine-tuning in macromolecular optimization (Igashov et al., 2023, Zhu et al., 4 Nov 2024, Rong et al., 11 Jun 2025).

7. Bridge Property, Memorylessness, and Pinning Point Analysis

The Markov or memoryless property of a bridge process depends critically on the law of the pinning point (the target endpoint).

• Discrete or singular measures: If the pinning point law has no absolutely continuous component with respect to Lebesgue measure, the Lévy bridge (with random length and random pinning) remains Markovian. Conditional expectations depend only on the current state, retaining the classical memoryless property (Louriki, 13 Jul 2024).
• Absolutely continuous pinning: When the pinning point’s law is absolutely continuous, the Markov property fails; the future of the process depends on more than the current state, losing the memoryless character (Louriki, 13 Jul 2024).
• This analysis suggests that preserving sharp Markov properties in discrete Markov bridges (e.g., for simulation or inference) is linked to using discrete or singular endpoint distributions—an insight with implications for tractability and modeling in finance, probability, and applied data science.

Table: Summary of Key Discrete Markov Bridge Themes

Principle or Application	Key Reference	Main Result/Usage
Doob $h$-transform for bridges	(Çetin et al., 2014)	Universal mechanism for conditioning on endpoints
Reciprocal invariants / classes	(Conforti et al., 2014, Conforti, 2015)	Uniqueness, duality formulas, bridge structure
Quantitative jump properties	(Conforti, 2015, Conforti, 2016)	Convexity, marginals, concentration, large deviations
Generative model frameworks	(Li et al., 26 May 2025, Pham et al., 11 Feb 2025)	Latent learning, score-based reversal, efficient parameterization
Simulation and inference	(Baltazar-Larios et al., 2023, Treut et al., 2023)	Fast sampling, MCEM, rare-event path sampling
Protein/chemical sequence design	(Igashov et al., 2023, Zhu et al., 4 Nov 2024, Rong et al., 11 Jun 2025)	Sequence–structure bridges, energy-based design
Markov property and pinning law	(Louriki, 13 Jul 2024)	Preservation or loss depends on endpoint distribution

Discrete Markov bridges, viewed as endpoint-conditioned Markov processes and operationalized through explicit $h$-transforms, reciprocal invariants, and learnable jump dynamics, constitute a versatile mathematical framework underpinning both classic stochastic modeling and emerging techniques in generative machine learning and complex systems analysis.