The Minimum Description Length (MDL) principle is a framework for model selection and statistical inference that uses data compression and coding theory. It posits that the best model for a set of data is the one that allows the shortest possible description of the data. The principle is grounded in the idea that the optimal code for data should capture its statistical characteristics, and that the length of the code reflects the complexity of the model. The MDL principle is closely related to the concept of stochastic complexity, which is the minimum description length of data under a given model class. The paper reviews the principles of MDL and stochastic complexity, showing that various coding methods, such as normalized maximum likelihood, mixture, and predictive codings, achieve stochastic complexity up to asymptotically vanishing terms. It discusses the performance of the MDL criterion in data compression and statistical inference, using examples like context tree modeling, density estimation, and model selection in Gaussian linear regression. The paper also explores the connection between MDL and algorithmic complexity, noting that MDL provides a global maximum likelihood principle. The MDL principle is particularly useful in the middle ground between parametric and nonparametric inference, where a variety of plausible model classes are available. It allows for the automated selection of an estimate based on data, balancing likelihood and parametric complexity. The paper also discusses the asymptotic equivalence of optimal solutions in average and worst-case scenarios, showing that the stochastic complexity of data relative to a model class is closely related to the parametric complexity. The paper further explores the strong optimality of stochastic complexity, demonstrating that it is optimal for most models in the sense of a theorem that generalizes Shannon's noiseless coding theorem. It also discusses the simplification of computations via sufficiency, showing that the NML distribution for sufficient statistics can be decomposed into the complexity of the statistic and the Shannon codelength for the data given the statistic. The paper concludes by highlighting the applications of the MDL principle in universal coding, linear regression, and density estimation, emphasizing its role in providing a unified framework for statistical modeling and inference.The Minimum Description Length (MDL) principle is a framework for model selection and statistical inference that uses data compression and coding theory. It posits that the best model for a set of data is the one that allows the shortest possible description of the data. The principle is grounded in the idea that the optimal code for data should capture its statistical characteristics, and that the length of the code reflects the complexity of the model. The MDL principle is closely related to the concept of stochastic complexity, which is the minimum description length of data under a given model class. The paper reviews the principles of MDL and stochastic complexity, showing that various coding methods, such as normalized maximum likelihood, mixture, and predictive codings, achieve stochastic complexity up to asymptotically vanishing terms. It discusses the performance of the MDL criterion in data compression and statistical inference, using examples like context tree modeling, density estimation, and model selection in Gaussian linear regression. The paper also explores the connection between MDL and algorithmic complexity, noting that MDL provides a global maximum likelihood principle. The MDL principle is particularly useful in the middle ground between parametric and nonparametric inference, where a variety of plausible model classes are available. It allows for the automated selection of an estimate based on data, balancing likelihood and parametric complexity. The paper also discusses the asymptotic equivalence of optimal solutions in average and worst-case scenarios, showing that the stochastic complexity of data relative to a model class is closely related to the parametric complexity. The paper further explores the strong optimality of stochastic complexity, demonstrating that it is optimal for most models in the sense of a theorem that generalizes Shannon's noiseless coding theorem. It also discusses the simplification of computations via sufficiency, showing that the NML distribution for sufficient statistics can be decomposed into the complexity of the statistic and the Shannon codelength for the data given the statistic. The paper concludes by highlighting the applications of the MDL principle in universal coding, linear regression, and density estimation, emphasizing its role in providing a unified framework for statistical modeling and inference.