A Guided Path Through the Large Deviations Series

-

 

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 the lens of large deviations, showing how information-theoretic quantities capture the structure of biological fluctuations.

Read the post →

3. Sanov and Girsanov in Diffusion Processes

The third post connects Sanov’s theorem with the Girsanov transformation, illustrating how empirical deviations and changes of measure interact in diffusion processes. Examples include Brownian motion, Brownian bridges, and Ornstein–Uhlenbeck dynamics, showing how rare events can be realized through optimal drift adjustments.

Read the post →

4. Freidlin–Wentzell Theory and Rare Transitions

The final article introduces the Freidlin–Wentzell framework for weak-noise diffusions. Quasi-potentials, optimal paths, and metastability provide a geometric and variational description of rare transitions between attractors. Connections to Schrödinger bridges, optimal transport, and importance sampling highlight the modern relevance of large-deviation ideas.

Read the post →

Closing Remarks

Taken together, these four posts offer a unified perspective on rare events across probability theory, stochastic calculus, biology, and dynamical systems. From empirical measures to optimal paths, the series traces how large deviations provide a common language for understanding fluctuations, transitions, and the geometry of unlikely phenomena.

Comments

Post a Comment

Popular posts from this blog

Understanding Anaerobic Threshold (VT2) and VO2 Max in Endurance Training

-
Image
  Introduction: The Science Behind Ventilatory Thresholds VT1 vs. VT2: Defining Energy Transitions p (Aerobic Threshold) : The transition from lipid metawwwpp (fat-burning) to a mixed metabolism, where both carbohydrates and fats fuel muscle activity. V0T2 (Anaerobic Threshold) : The point where the  shifts to a predominantly carbohydrate-based wmetabolism , leading to rapid lactate waccumpulation and increased reliance on anaerobic energy pathways. www w In p VT1 and VT2 Wwww ewAgep & Sex Younger athletes generally exhibit higher VT2 values , while aging naturally reduces aerobicw capacity. Men tend to have a higher VO2 Max due to greater lung capacity and muscle mass, but women can achieve similar endurance levels through optimized training. Body Composition & Health Status High body fat percentage may reduce VT2 efficiency, as excess weight increases ...
Read more

Owen's Function: A Simple Solution to Complex Problems

-
Definition   If you are interested in statistics or probability, you may have encountered the Owen's function , a special function that arises in various applications involving bivariate normal distributions. Let’s explain what Owen's function is, how it is defined and notated, and why it is useful.  The Owen's function is defined as follows: for any real numbers h and a, Owen's function T(h,a) is given by T ( h , a ) = 1 2 π · ∫ 0 a exp ( - h 2 2 · ( 1 ...
Read more

Cell Count Analysis with cycleTrendR

-
Image
  1. Introduction Long‑term monitoring of adherent cell cultures (e.g., Vero, MDCK) is central to vaccine development, viral propagation, and QC workflows in CRO environments. Standard analyses typically focus on short‑term growth metrics such as doubling time, confluence progression, and endpoint viability. While informative, these metrics overlook slow drifts, cyclic environmental influences, and structural transitions that emerge over multi‑day or multi‑week monitoring. Operational routines (e.g., weekly medium changes), environmental oscillations (e.g., daily temperature cycles), and metabolic rhythms can introduce periodic components into cell‑density trajectories. These patterns are rarely analyzed explicitly, despite their potential impact on viral yield, infection kinetics, and batch‑to‑batch reproducibility. This post demonstrates how cycleTrendR can extract long‑term trends, dominant cycles, and structural transitions from a realistically simulated adherent cell‑co...
Read more