Part II: The Likelihood Catalogue

Part II serves as a comprehensive reference catalogue of likelihood functions encountered across statistics and data science. For each distribution we derive the likelihood, log-likelihood, score function, Fisher information, and maximum-likelihood estimator from first principles, showing every algebraic step so the reader can reproduce each result with pencil and paper.

The catalogue is organised into four chapters:

  • Chapter 5 (Chapter 5 — Discrete Likelihoods) covers discrete distributions whose support is a countable set of integers: Bernoulli, Binomial, Poisson, Negative Binomial, Geometric, Hypergeometric, Multinomial, and the Zero-Inflated Poisson.

  • Chapter 6 (Chapter 6 — Continuous Likelihoods) covers continuous distributions defined by a probability density function on the real line (or a subset thereof): Normal, Exponential, Gamma, Beta, Log-Normal, Weibull, Pareto, Student-t, Chi-squared, F, Uniform, and Cauchy.

  • Chapter 7 (Chapter 7 — Multivariate Likelihoods) extends the treatment to vector-valued and matrix-valued distributions—Multivariate Normal, Wishart, Inverse-Wishart, Dirichlet, and Multinomial—introducing the matrix-calculus tools required for their derivations.

  • Chapter 8 (Chapter 8 — Specialized Likelihoods) surveys likelihood variants that arise when the standard i.i.d. likelihood is unavailable or inconvenient: profile, partial, marginal, conditional, composite, quasi-, pseudo-, and empirical likelihoods, together with censored/truncated likelihoods and penalised likelihoods.

Likelihood Catalogue