Conditional splitting probabilities for hidden-state inference in drift-diffusive processes

Technical Summary: Conditional Splitting Probabilities for Hidden-State Inference in Drift-Diffusive Processes

Problem Statement
Splitting probabilities are fundamental quantities in stochastic processes, quantifying the likelihood of a system exiting a defined interval through specific boundaries. While traditionally applied to one-dimensional processes to determine exit probabilities (e.g., Gambler's ruin, fixation in population dynamics), this work addresses a more complex scenario involving two-dimensional Markov processes $\{X(t), Y(t)\}$. Here, $X(t)$ is an observable, continuous drift-diffusive variable, while $Y(t)$ represents a hidden internal state (which may be discrete or continuous). The central problem is to determine the joint splitting probability: the likelihood that $X(t)$ exits an interval $L = [x_{min}, x_{max}]$ through a specific boundary and that the hidden state $Y(t)$ is in a specific configuration $y_{exit}$ at the moment of exit.

Building on this, the paper investigates the conditional splitting probability: the posterior likelihood of the hidden state $Y$ given that an exit event has occurred at a specific boundary. The authors aim to determine if and how the observation of a single point-wise exit event can be used to infer the state of a hidden variable, a capability relevant to active matter control and decision theory.

Methodology
The authors employ a framework based on the Kolmogorov backward equation and Bayes' theorem.

1. Joint Probability Formulation:
   For a process governed by forward Fokker-Planck operators $\mathcal{L}^{(X)}_F$ and $\mathcal{L}^{(Y)}_F$, the joint splitting probability $\Pi(x_{exit}, y_{exit}|x_0, y_0)$ satisfies the stationary backward equation:
   $$0 = \mathcal{L}^{(X)}_B \Pi + \mathcal{L}^{(Y)}_B \Pi$$
   subject to boundary conditions where the process is absorbed at the interval boundaries. The authors consider both fully permeable (absorbing) and partially permeable (Robin) boundary conditions.

2. Decoupled vs. Coupled Dynamics:
   The analysis is divided into two classes of processes:

   • Decoupled Case: $X(t)$ is a simple Brownian motion independent of $Y(t)$. The solution is constructed using the eigenbasis of the Fokker-Planck operator for $Y(t)$. The joint probability is expressed as a spectral expansion involving the eigenfunctions and eigenvalues of the hidden process.
   • Unidirectionally Coupled Case: $X(t)$ depends on $Y(t)$ (e.g., drift or diffusion coefficients depend on the internal state), but $Y(t)$ evolves independently of $X(t)$. In these cases, the backward equations become coupled systems of ODEs, which are solved explicitly for specific paradigmatic models.

3. Conditional Inference:
   Once the joint probabilities are computed, the conditional splitting probability is derived via Bayes' theorem:
   $$\Pi(y_{exit}|x_{exit}; x_0, y_0) = \frac{\Pi(x_{exit}, y_{exit}|x_0, y_0)}{P(x_{exit}|x_0, y_0)}$$
   where the denominator is the marginal splitting probability of the observable $X$.

Key Results

1. Decoupled Hidden States (Brownian Motion + Internal State)
For processes where $X(t)$ is Brownian motion and $Y(t)$ is a decoupled internal state (discrete or continuous), the authors derive generic expressions for the joint splitting probability in terms of the eigensystem of the $Y$-dynamics.

• Ripening-Spoiling Process: A discrete three-state model (Unripe $\to$ Ripe $\to$ Spoiled) is analyzed. The joint probability of exiting in a specific state (e.g., "Ripe") exhibits non-monotonic dependence on the initial position $x_0$. This arises because only trajectories with specific durations (allowing transition to "Ripe" but not yet to "Spoiled") contribute to the exit probability in that state.
• Ornstein-Uhlenbeck (OU) Process: For a continuous Gaussian hidden state, the joint probability is expressed as an infinite series of Hermite polynomials.
• Inference Limit: In the decoupled case, the authors demonstrate that inference of the hidden state from a single exit event is only possible in the transient regime (non-stationary initial conditions). In the limit of large intervals ($L \to \infty$) or vanishing diffusion ($D \to 0$), the joint probability factorizes into the product of the marginal splitting probability and the steady-state distribution of $Y$. Consequently, the conditional probability converges to the prior, rendering inference impossible for stationary initializations.

2. Unidirectionally Coupled Hidden States
For systems where the observable dynamics depend on the hidden state, inference remains possible even for stationary initializations. The authors provide explicit derivations for three cases:

• Run-and-Tumble (RnT) Dynamics: A particle with a self-propulsion state $Y \in \{-1, 1\}$ switching at rate $\alpha$. The authors solve the coupled backward equations to find joint splitting probabilities. They show that the posterior likelihood of the particle's direction (left vs. right mover) upon exiting a boundary deviates significantly from the prior (1/2). This deviation increases with the Péclet number ($Pe = \nu^2/\alpha D$), indicating that persistence allows for reliable inference of the internal state even in the limit of large intervals.
• Intermittent Piecewise-Linear Potential: A particle subject to a potential that switches "on" and "off" stochastically. The joint probabilities are solved by splitting the domain at the potential apex. The results show that the conditional probability of the potential being "on" or "off" at exit depends non-monotonically on the apex location and barrier height.
• Stochastic Resetting: A particle undergoing Brownian motion with stochastic resetting to a position $x_r$. The hidden state is defined as the number of reset events $n$. The authors derive the joint probability of exiting after exactly $n$ resets. They find that while the most likely number of resets is zero (due to the memoryless nature of the Poisson process), the average number of resets conditioned on a specific exit boundary is non-trivial and depends on the resetting rate and interval size.

Significance and Claims
The paper claims two primary contributions:

1. Analytical Framework: It provides a general method to compute joint splitting probabilities for two-dimensional Markov processes, expressed either via spectral decomposition (for decoupled systems) or explicit solutions to coupled ODEs (for unidirectionally coupled systems).
2. Inference Scheme: It establishes a "simple scheme" for the partial Bayesian inference of hidden internal states based solely on point-wise detection of exit events.

The authors highlight a critical distinction between the two classes of processes:

• In decoupled systems, inference is limited to transient dynamics; once the hidden state relaxes to its steady state, exit events provide no information about the internal state.
• In unidirectionally coupled systems (common in active matter models), the coupling preserves information about the internal state in the exit statistics, allowing for inference even in the long-time limit ($L \to \infty$).

The work suggests that this boundary-based inference scheme could be applied to the control of driven stochastic processes, such as optimizing power extraction from active particles, by leveraging the correlation between exit points and hidden states. The authors note that while they have focused on discrete hidden states in the coupled cases, extending these results to continuous hidden states (e.g., active Ornstein-Uhlenbeck processes) remains an open challenge.