Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm.Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propagation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible.Stan can be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces support sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting.

Stan is a free and open-source Cþþ program that performs Bayesian inference or optimization for arbitrary user-specified models and can be called from the command line, R, Python, Matlab, or Julia and has great promise for fitting large and complex statistical models in many areas of application. We discuss Stan from users' and developers' perspectives and illustrate with a simple but nontrivial nonlinear regression example.

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Because of x-ray dose considerations or practical mechanical restrictions, it may be interesting for some applications to perform limited-angle scanning. So the limitedangle problem is of practical significance in computed tomography (CT). Furthermore, it is an ill-posed problem. Total variation minimization (TVM) and projection on convex sets (POCS) based iterative reconstruction algorithm is comparatively valid for this undersampling CT reconstruction problem and can obtain comparatively good performance. But the reconstruction images have artifacts with gradual change gray nearby edges. In this paper, a simple but efficient method to eliminate such artifacts during the reconstruction is proposed. Based on the same assumption of the TVM algorithm, for the object whose density distribution is approximate piecewise-constant, we develop and investigate an improved iterative reconstruction algorithm for volume image reconstruction from limited-angle cone-beam scan, which is referred to as piecewise-constant modification TVM-POCS (PM-TVM-POCS) algorithm. The grays of voxels with gradually changed gray artifacts are modified in the TVM-POCS iteration process by the piecewiseconstant modification algorithm which is regarded as a "collapse" process, and gradually approaches the true piecewise-constant image. The results of simulation experiments show that the presented modification algorithm can improve the quality of the reconstructed image obviously and it is steady to noise. This modification algorithm can also be applied to other reconstruction problems which have artifacts with gradual change gray.