LabStats

  This post serves as a short guide to the four-part series on large deviations and their applications to stochastic processes, biology, and weak-noise dynamical systems. Each article can be read independently, but together they form a coherent narrative that moves from foundational principles to modern applications. 1. Sanov’s Theorem and the Geometry of Rare Events The series begins with an intuitive introduction to Sanov’s theorem , highlighting how empirical distributions deviate from their expected behavior and how the Kullback-Leibler divergence emerges as the natural rate functional. This post lays the conceptual groundwork for understanding rare events in high-dimensional systems. Read the post → 2. Sanov’s Theorem in Living Systems The second article explores how Sanov’s theorem applies to biological and neural systems . Empirical measures, population variability, and rare transitions in gene expression or neural activity are framed through ...

  After introducing Sanov’s theorem , Girsanov transformations , and their applications to diffusion processes, we now turn to the broader framework provided by Freidlin–Wentzell theory . This theory describes how stochastic systems with weak noise escape from basins of attraction, offering a geometric and variational perspective on rare transitions, metastability, and escape times. Notation For clarity, we summarize the variables and symbols used throughout this post: \(X_t^\varepsilon\) : diffusion process with small noise intensity \(\varepsilon\). \(\varepsilon\) : noise amplitude, assumed to satisfy \(0 \(b(x)\) : deterministic drift field defining the noiseless dynamics \(\dot{x} = b(x)\). \(\sigma\) : diffusion coefficient (scalar or matrix), assumed non-degenerate. \(W_t\) : standard Brownian motion. \(\phi(t)\) : candidate path used in the variational formulation of large deviations. \(\dot{\phi}(t)\) : time derivative of the path \(\phi(t...

  Large-deviation theory and change-of-measure techniques provide a powerful toolkit for analyzing rare events in stochastic processes. In this post, we explore how Sanov’s theorem and the Girsanov transformation interact in the context of two fundamental diffusion models: Brownian motion and the Ornstein–Uhlenbeck process. 1. Introduction Sanov’s theorem describes the exponential decay of probabilities associated with atypical empirical distributions of i.i.d. random variables. Girsanov’s theorem, on the other hand, provides a way to modify the drift of a stochastic process by changing the underlying probability measure. Together, they form a natural bridge between empirical deviations and pathwise deviations in diffusion processes. Notation For clarity, we summarize the variables and symbols used throughout this post: \(X_t\) : generic diffusion process. \(B_t\) : Brownian motion with volatility \(\sigma\). \(\sigma\) : diffusion coefficient...

Biological and neural systems operate under substantial intrinsic stochasticity. Despite this, they maintain stable distributions of molecular, cellular, and population-level states. Occasionally, however, these systems exhibit rare, high-impact deviations. Sanov’s theorem provides a unifying framework for quantifying the improbability of such events through the geometry of empirical distributions. 1. Introduction Rare events in living systems often correspond not to isolated fluctuations but to atypical empirical distributions emerging from many interacting components. Sanov’s theorem characterizes the exponential decay of the probability of such deviations using the Kullback–Leibler divergence. This post surveys applications across molecular biology, cell dynamics, and neuroscience. Notation The following symbols and variables are used throughout this post: \(X_1, \dots, X_n\) : observations or molecular states sampled from a biological system. \(P\) : bas...

Large-deviation theory provides a powerful mathematical framework for quantifying the probability of rare events in stochastic systems. Among its foundational results, Sanov’s theorem plays a central role: it characterizes the exponential decay of probabilities associated with atypical empirical distributions. This perspective is deeply connected to information theory, statistical mechanics, and the analysis of stochastic processes. 1. Introduction Many stochastic systems—ranging from molecular dynamics to neural activity—exhibit fluctuations that are typically small but occasionally produce rare, high-impact deviations. Understanding the probability of these deviations requires tools that go beyond classical variance-based approximations. Sanov’s theorem provides exactly such a tool: a geometric and information-theoretic description of rarity. This post introduces the theorem, its intuition, and its connection to diffusion processes through discrete approximations and con...

  A follow‑up to “Cell Count Analysis with cycleTrendR” — focused on neurodegenerative drift In our previous post , we explored how cycleTrendR can be applied to cell‑count trajectories in longitudinal culture experiments. That example sparked an insightful question on LinkedIn : How does cycleTrendR perform when the signal is slow, noisy, and stretched across long timelines — as in neurodegenerative disease models? This post is a direct response. Neurodegenerative phenotypes often evolve gradually, with subtle changes accumulating over weeks or months. Meanwhile, experimental noise — imaging variability, segmentation artefacts, media‑change cycles — fluctuates rapidly and can dominate the signal. cycleTrendR was designed precisely for this challenge: to extract slow biological drift from noisy, irregular, and cyclic time series. How cycleTrendR isolates slow biological drift from fast experimental noise Longitudinal cell‑culture experiments — especially those modelling neurodege...