• bayesian
• hamiltonian monte carlo
• inference
• lisa
• markov chain
• monte carlo
• regression
• sampler

Hamiltonian Monte Carlo (HMC) are a special class of Markov Chain Monte Carlo algorithms (MCMC) which demonstrated empirical success in many foregrounds due to their ability to efficiently sample in the typical set, even for high dimensions problem.
The parameters space is embedded with auxiliary momentum variables to generate a fake Hamiltonian where the potential energy is proportional to the target probability distribution. In this way, the information about the parameter space is taken into account for new efficient proposal. The Hamiltonian Dynamics are evolved at each step of the Markov chain, to ensure a transition to state space at low autocorrelation and avoiding the tedious random walk behaviour of many MCMC samplers. This result in much more efficiency and less waste of computational power.
This thesis delve to study the theoretical foundation of Hamiltonian Monte Carlo algorithm in its simpler form and its variations and improvements. The implementation is carried out in python and comprehend the realization of the existing algorithms as well as the presentation of a novel version. Specifically, Metropolis-Hastings and Metropolis-Adjested Langevin Algorithm (MALA), have been confronted to HMCs samplers of various nature.
The Gaussian distribution and Rosenbrock function have been used as testing ground to ensure the validity of the algorithms as well as demonstrate their power. However, more sophisticated approaches are employed to validate their statistical behavior. Instead of testing a single realization of the MCMCs, automatic validation software is used to ensure the statistical correctness of the algorithms. Using a pp-plot, the posterior quantiles across many runs are analyzed and compared with the expected behavior.
To further illustrate the capabilities of HMCs, other significant benchmarks have been employed. Using a Bayesian approach, linear regression and logistic regression problems have been addressed with MCMC algorithms, and their results have been compared.
Finally HMCs algorithms are tested to analyze their suitability to infer parameters from Gravitational Waves (GWs) signals captured by the Laser Interferometer Space Antenna (LISA). LISA consists of three spacecraft arranged in a triangular formation, each separated by 2.5 million kilometers, orbiting the Sun in a precise and stable formation. Utilizing laser interferometry, LISA will measure the minute changes in distance between the spacecraft caused by passing gravitational waves. LISA aims to detect and measure the GWs caused by some of the most violent and energetic processes in the cosmos, such as the mergers of supermassive black holes, the inspiral of compact binary systems, and possibly signals from the early universe itself.