Book: Analytical Theory of Probabilities

Overview
Pierre-Simon Laplace's Théorie analytique des probabilités (1812), translated as Analytical Theory of Probabilities, presents a systematic, mathematically rigorous foundation for probability and its applications. It treats probability as an analytic discipline, using calculus, infinite series, and integral approximations to derive general laws and practical formulas. The work combines formal derivations with extensive applied examples drawn from astronomy, demographic records, and legal and moral questions.
Laplace frames probability as both a mathematical science and a tool for reasoning under uncertainty. Emphasis falls on methods that allow one to pass from combinatorial problems to continuous approximations, and on techniques that turn otherwise intractable sums and integrals into usable asymptotic results. The text is notable for its breadth, uniting pure analytic development with concrete problems of data and inference.

Analytical methods and generating functions
A central theme is the use of analytic machinery to handle sums and distributions. Generating functions appear as a powerful device for encoding probability laws and extracting coefficients that describe discrete distributions. Laplace also develops integral approximations and what came to be known as "Laplace's method" for approximating the value of integrals in the limit of large parameters, a tool that underpins many asymptotic results.
The analytic viewpoint allows Laplace to derive continuous approximations to discrete phenomena, to justify the emergence of the normal curve as an approximation to sums of independent effects, and to present systematic procedures for asymptotic expansion. This interplay between series, integrals, and approximation techniques shaped later formal tools in probability theory.

Probability, inference, and the rule of succession
Laplace gives a thorough account of "inverse probability," now understood as Bayesian inference, using prior beliefs and observed data to update probabilities for unknown quantities. He formalizes procedures for computing posterior distributions, point estimates, and intervals of confidence based on observed evidence. His exposition includes the famous "rule of succession" for predicting future events on the basis of past occurrences, an early and influential statement of Bayesian predictive reasoning.
His approach typically employs simple, often uniform, priors and then integrates analytically to obtain posterior probabilities, illustrating how prior information and data combine. This strand of the book links philosophical questions about probability to concrete computational recipes and stimulated much later debate and development in statistical inference.

Theory of errors and applications
Practical problems of measurement and observational error receive sustained attention. Laplace analyzes the distribution of measurement errors, derives formulas for combining observations, and provides justification for using averages and least-squares-like procedures for improving estimates. Applications include the processing of astronomical observations, estimation of population parameters from censuses, and probability assessments in legal testimony and insurance contexts.
He treats real data problems with the same analytic rigor, showing how probabilistic reasoning can produce actionable numerical estimates and error bounds. This applied focus demonstrated the utility of probability for science and public affairs, and provided templates for statisticians working with noisy measurements.

Legacy and influence
The Analytical Theory of Probabilities had profound influence on the development of both probability theory and statistical practice. Laplace's analytic techniques, asymptotic approximations, and advocacy of inverse probability shaped later formalizations of the central limit theorem, likelihood-based methods, and Bayesian thought. His blend of rigorous mathematics and concrete application set a standard for mathematical statistics in the 19th century.
Many methods and ideas from the book, approximation methods for integrals, analytic handling of generating functions, and Bayesian updating, remain central to modern probability and statistics. The work stands as a cornerstone in the transition from combinatorial probability to a continuous, analytic science of uncertainty.