Bayesian Networks and Graphical Models
Real problems have many interacting variables. Bayesian networks (directed graphical models) make such joint distributions tractable by drawing the dependencies as a graph: an arrow means “directly influences,” a missing
阅读文章 →Computational Bayes: MCMC and Variational Inference
Conjugacy is a luxury; most real posteriors have no closed form because the evidence integral p(D)= p(D )p( ) d is intractable. The modern Bayesian toolkit sidesteps it two ways: MCMC draws samples from the posterior wit
阅读文章 →Bayesian Regression and Hierarchical Models
Now we put the machinery to work on the models practitioners use every day. Bayesian regression replaces point-estimate coefficients with full posteriors, delivering principled uncertainty on every prediction. Hierarchic
阅读文章 →Model Comparison, Evidence, and Occam's Razor
Inference within a model assumes the model is right. But which model should we use? Bayesian model comparison answers this with the very quantity we worked so hard to avoid — the evidence p(D) — which turns out to automa
阅读文章 →Bayes in Machine Learning, Deep Learning, and AI
This capstone gathers the threads into one claim: Bayesian reasoning is the theory of learning under uncertainty, and it quietly underwrites much of machine learning. The loss is a likelihood. Regularization is a prior.
阅读文章 →Notation and Formula Reference
A compact reference for the symbols, rules, and formulas used throughout the book.
阅读文章 →Functions and the Idea of Calculus
Calculus studies how things change and how things accumulate. Before we can speak of change, we need the object that changes: the function. This chapter sets up functions and the handful of “essential” ones that appear e
阅读文章 →Limits and Continuity
The limit is the idea that makes calculus rigorous. “Instantaneous” rate of change and “infinitely thin” rectangles both hide a limit: a quantity we approach but may never reach. This chapter builds limits intuitively an
阅读文章 →The Derivative
The derivative is the central object of differential calculus and the single most important concept for machine learning. It answers the tangent problem: the instantaneous rate at which a function changes. Once we have i
阅读文章 →Differentiation Rules
Computing every derivative from the limit definition would be exhausting. Fortunately, a small set of rules lets us differentiate almost any function mechanically. The most important of these — the chain rule — is the ma
阅读文章 →Applications of the Derivative
Now that we can compute derivatives, we put them to work. Derivatives find the highest and lowest points of a function (optimization), approximate complicated functions by simple ones (linearization and Newton's method),
阅读文章 →The Integral and the Fundamental Theorem
Integration is the second pillar of calculus: the mathematics of accumulation. Where the derivative chops a curve into instantaneous rates, the integral sums infinitely many tiny pieces into a total — an area, a distance
阅读文章 →Techniques of Integration
Differentiation is mechanical; integration is an art. There is no universal algorithm, so we build a toolkit of techniques — each one reversing a differentiation rule — plus numerical methods for the (many) integrals wit
阅读文章 →Applications of Integration
Integration is far more than “area under a curve.” Any quantity built by accumulating infinitely many infinitesimal pieces is an integral: areas between curves, volumes of solids, lengths of arcs, averages, and — most im
阅读文章 →Sequences, Series, and Taylor Series
Can you add up infinitely many numbers and get a finite total? Sometimes — and when you can, the result is one of the most powerful tools in mathematics: the ability to represent complicated functions as infinite polynom
阅读文章 →Multivariable Calculus: Partial Derivatives and the Gradient
Machine-learning models depend on millions of parameters at once, so single-variable calculus is not enough. This chapter extends differentiation to functions of several variables. The star object is the gradient — the v
阅读文章 →Multiple Integrals and a Glimpse of Vector Calculus
Just as differentiation extended to many variables, so does integration. Multiple integrals accumulate a quantity over a region of two, three, or more dimensions — the natural setting for probabilities over several varia
阅读文章 →Optimization and Differential Equations
This chapter gathers the calculus that most directly becomes machine learning: how to find minima of functions of many variables (with and without constraints), the gradient-descent dynamics that training uses, and the d
阅读文章 →Calculus in Machine Learning, Deep Learning, and AI
Here is the payoff. Linear algebra gives AI its vocabulary — vectors, matrices, transformations. Calculus gives it the ability to learn. At the heart of almost every modern machine-learning system is one computational pa
阅读文章 →Calculus — Preface: how to use this book
If linear algebra is the language of machine learning, calculus is its engine of change. Calculus is the mathematics of things that vary — how a quantity responds to a small nudge in another (the derivative), and how cou
阅读文章 →Notation and Formula Reference
A compact reference for the symbols and identities used throughout the book.
阅读文章 →Vectors and the Geometry of Space
Everything in linear algebra is built from one object: the vector. Before we can multiply matrices or diagonalize transformations, we need to be completely comfortable with what a vector is, how to combine vectors, and h
阅读文章 →Systems of Linear Equations
A system of linear equations asks: which values of the unknowns satisfy several linear constraints at once? This is the oldest problem in the subject and still its computational core — training a linear model, balancing
阅读文章 →Matrices and Matrix Algebra
A matrix is the workhorse object of linear algebra. This chapter develops the algebra of matrices — how to add, multiply, transpose, and invert them — always keeping sight of the three views from the preface: a matrix is
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