I’ll describe Stan’s probabilistic programming language, and how it’s used, including

• blocks for data, parameter, and predictive quantities • transforms of constrained parameters to unconstrained spaces, with automatic Jacobian corrections • automatic computation of first- and higher-order derivatives • operator, function, and linear algebra library • vectorized density functions, cumulative distributions, and random number generators • user-defined functions • differential equation solvers

I’ll also provide an overview of the underlying algorithms for log density and derivative calculations, and inference with full Bayes, approximate Bayes, and maximum likelihood:

• adaptive Hamiltonian Monte Carlo for Markov-chain Monte Carlo sampling • L-BFGS optimization and transforms for maximum likelihood estimation • Automatic differentiation variational inference (ADVI) for variational inference

I’ll also briefly describe the user-facing interfaces: RStan (R), PyStan (Python), CmdStan (command line), Stan.jl (Julia), MatlabStan (MATLAB), MathematicaStan (Mathematica)

I’ll finish with an overview of the what’s on the immediate horizon:

• GPU matrix operations • MPI multi-core, multi-machine parallelism • data parallel expectation propagation for approximate Bayes • marginal Laplace approximations