Browsing by Subject "Lagrangian Monte Carlo"

The focus of this work is to efficiently sample from a given target distribution using Monte Carlo Makov Chain (MCMC). This work presents No-U-Turn Sampler Lagrangian Monte Carlo with the Monge metric. It is an efficient MCMC sampler, with adaptive metric, fast computations and with no need to hand-tune the hyperparameters of the algorithm, since the parameters are automatically adapted by extending the No-U-Turn Sampler (NUTS) to Lagrangian Monte Carlo (LMC). This work begins by giving an introduction of differential geometry concepts. The Monge metric is then constructed step by step, carefully derived from the theory of differential geometry giving a formulation that is not restricted to LMC, instead, it is applicable to any problem where a Riemannian metric of the target function comes into play. The main idea of the metric is that it naturally encodes the geometric properties given by the manifold constructed from the graph of the function when embedded in higher dimensional Euclidean space. Hamiltonian Monte Carlo (HMC) and LMC are MCMC samplers that work on differential geometry manifolds. We introduce the LMC sampler as an alternative to Hamiltonian Monte Carlo (HMC). HMC assumes that the metric structure of the manifold encoded in the Riemannian metric to stay constant, whereas LMC allows the metric to vary dependent on position, thus, being able to sample from regions of the target distribution which are problematic to HMC. The choice of metric affects the running time of LMC, by including the Monge metric into LMC the algorithm becomes computationally faster. By generalizing the No-U-Turn Sampler to LMC, we build the NUTS-LMC algorithm. The resulting algorithm is able to estimate the hyperparameters automatically. The NUTS algorithm is constructed with a distance based stopping criterion, which can be replaced by another stopping criteria. Additionally, we run LMC-Monge and NUTS-LMC for a series of traditionally challenging target distributions comparing the results with HMC and NUTS-HMC. The main contribution of this work is the extension of NUTS to generalized NUTS, which is applicable to LMC. It is found that LMC with Monge explores regions of target distribution which HMC is unable to. Furthermore, generalized NUTS eliminates the need to choose the hyperparameters. NUTS-LMC makes the sampler ready to use for scientific applications since the only need is to specify a twice differentiable target function, thus, making it user friendly for someone who does not wish to know the theoretical and technical details beneath the sampler.