Variational Laplace for Dummies#
Welcome to Variational Laplace for dummies: A Journey from Zero to Understanding. This book is born out of a personal journey—a quest to understand complex statistical methods without the heavy mathematical background often assumed by experts in the field.
As a biologist with a PhD in consciousness neuroscience, my academic path included minimal formal training in advanced mathematics. My experience with statistics was rooted in practical applications: from simple t-tests to more complex linear models, and incorporating machine learning and multivariate methods into my research. However, as I delved deeper into the nuances of scientific inquiry, I found myself drawn to Bayesian statistics.
Bayesian methods appealed to me for their flexibility and their ability to provide a formal quantification of evidence. They align closely with the kinds of questions that drive scientific exploration: “How much evidence do I have for a particular hypothesis or theory?” Specifically, I became interested in the Variational Laplace method. It offers a way to compute the model evidence component of Bayes’ theorem for a wide array of models used in neuroscience, including univariate and multivariate generalized linear models, whether mixed or not.
However, I quickly felt overwhelmed when reading papers and textbook explaining variational laplace. The existing papers and resources on Variational Laplace were dense, full of what looked like complex mathematics, and often assumed a level of prior knowledge I did not possess. It wasn’t that the authors failed to explain the concepts, but rather that certain connections that were obvious to them and their target readers were lost on me. I ended up confused, mixing up concepts, not sure which symbol refers to what and so on. You probably know the feeling.
I was however really determined to understand it, because I wanted to write papers leveraging this method quite heavily to answer questions I couldn’t otherwise. To my surprise, I discovered that the core concepts could be grasped using the basic algebra and arithmetic I learned in high school. The real challenge was not the complexity of the mathematics itself but the way it was presented—by people better versed in mathematics, for people better versed in mathematic than I am.
That’s why I wrote this book: variational Laplace by a dummy, for dummies (no offense intended). My goal is to provide a clear, accessible introduction to Variational Laplace for statistical modeling and inference, aimed at readers who, like me, may not have an extensive mathematical background. We will start with the simplest generalized linear model: a linear model with a single regressor. From there, we’ll gradually build up to more complex models, adding additional regressors and eventually exploring multivariate modeling.
Throughout the book, I’ve employed several strategies to make the material as understandable as possible:
Detailed Explanations: I explain concepts thoroughly, sometimes repeating ideas in different ways to reinforce understanding.
Mathematical Formulas with Plain Language: Each formula is accompanied by a verbal explanation, clarifying how the mathematical expressions align with the concepts discussed.
Python Code Translations: Every significant formula is translated into Python code. This not only helps in visualizing the concepts but also makes the material accessible to those who may be more comfortable with programming than with mathematical notation.
Interactive Jupyter Notebooks: The content is presented using Jupyter notebooks, allowing for hands-on exploration. Readers can modify code, see immediate outputs, and engage with visualizations that reinforce the material
While the primary focus of this book is on Variational Laplace as a statistical tool, it’s worth noting its broader significance in neuroscience. The mathematics we explore here form the backbone of the Free Energy Principle (FEP) and frameworks like Predictive Processing, which are influential in modern neuroscience. By understanding the material in this book, you’ll also be better equipped to delve into these advanced topics, even though they are not our main focus. Consider it an added bonus—a “freebie”—that enhances the value of your learning experience.
It’s important to mention that this book is a work in progress. To ensure accuracy, I’ve collaborated with a co-author who is well-versed in mathematics. They’ve reviewed the content to confirm that it makes sense from a mathematical standpoint. Additionally, other colleagues have provided feedback to enhance clarity and correctness. However, as the material has not yet been fully verified and proofread by mathematicians, I encourage you to approach it with a critical mind and consider it a stepping stone rather than a definitive resource.
In writing this book, I aim to open doors for those who have felt intimidated or excluded by the complexity of advanced statistical methods. Whether you’re a fellow biologist, a neuroscientist, or simply someone curious about Bayesian statistics, I hope this book serves as a friendly guide on your journey to understanding Variational Laplace and its applications.
Let’s get started.