Stan is a probabilistic programming language for specifying statistical models. It allows for full Bayesian inference in continuous-variable models using Markov chain Monte Carlo (MCMC) methods like the No-U-Turn sampler (NUTS), and penalized maximum likelihood estimates using optimization methods such as the BFGS algorithm. Stan also provides tools for computing log densities and their gradients and Hessians, which are useful in algorithms like variational Bayes and expectation propagation. It can be used from the command line, R via RStan, or Python via PyStan. Stan's design allows for efficient sampling and optimization, with support for both MCMC and optimization-based inference. It includes a modeling language that enables statisticians to write models in familiar notation, which are then translated into efficient C++ code. The language supports imperative constructs like conditionals and loops, making it more flexible than BUGS. Stan addresses challenges in modeling by implementing efficient algorithms for constrained variables, such as positive variables, simplexes, and covariance matrices. It also includes an adaptive diagonal metric for HMC, which automatically standardizes parameter scales. The NUTS sampler is used for efficient sampling, and Stan supports Riemannian manifold Hamiltonian Monte Carlo (RHMC) for improved performance in hierarchical models. The paper describes a Bernoulli model example, showing how Stan can be used to estimate a parameter. It includes data formatting, model compilation, and sampling. The output includes summaries of the model, including posterior means, standard deviations, and convergence diagnostics. Stan also provides optimization for finding posterior modes, with results compared to MCMC sampling. Stan's diagnostic mode checks the log probability and gradient calculations at initial parameter values. The language includes blocks for data, transformed data, parameters, transformed parameters, model, and generated quantities, allowing for complex hierarchical models. It supports missing data models, with missing values declared as parameters. Stan's implicit uniform priors and non-centered parameterizations are discussed, with the latter being recommended for sparse data. Stan's design emphasizes efficiency and flexibility, with support for both MCMC and optimization. It provides tools for posterior inference, predictive checks, and efficient sampling, making it a powerful tool for statistical modeling and inference.Stan is a probabilistic programming language for specifying statistical models. It allows for full Bayesian inference in continuous-variable models using Markov chain Monte Carlo (MCMC) methods like the No-U-Turn sampler (NUTS), and penalized maximum likelihood estimates using optimization methods such as the BFGS algorithm. Stan also provides tools for computing log densities and their gradients and Hessians, which are useful in algorithms like variational Bayes and expectation propagation. It can be used from the command line, R via RStan, or Python via PyStan. Stan's design allows for efficient sampling and optimization, with support for both MCMC and optimization-based inference. It includes a modeling language that enables statisticians to write models in familiar notation, which are then translated into efficient C++ code. The language supports imperative constructs like conditionals and loops, making it more flexible than BUGS. Stan addresses challenges in modeling by implementing efficient algorithms for constrained variables, such as positive variables, simplexes, and covariance matrices. It also includes an adaptive diagonal metric for HMC, which automatically standardizes parameter scales. The NUTS sampler is used for efficient sampling, and Stan supports Riemannian manifold Hamiltonian Monte Carlo (RHMC) for improved performance in hierarchical models. The paper describes a Bernoulli model example, showing how Stan can be used to estimate a parameter. It includes data formatting, model compilation, and sampling. The output includes summaries of the model, including posterior means, standard deviations, and convergence diagnostics. Stan also provides optimization for finding posterior modes, with results compared to MCMC sampling. Stan's diagnostic mode checks the log probability and gradient calculations at initial parameter values. The language includes blocks for data, transformed data, parameters, transformed parameters, model, and generated quantities, allowing for complex hierarchical models. It supports missing data models, with missing values declared as parameters. Stan's implicit uniform priors and non-centered parameterizations are discussed, with the latter being recommended for sparse data. Stan's design emphasizes efficiency and flexibility, with support for both MCMC and optimization. It provides tools for posterior inference, predictive checks, and efficient sampling, making it a powerful tool for statistical modeling and inference.