The Standard Model (SM) of particle physics is a theory that describes the dynamics of subatomic particles. The Higgs particle is an essential component of the SM; its existence explains why other elementary particles are massive. The existence of the Higgs boson needs to be confirmed by experiment. The Large Hadron Collider (LHC) at the European organization for nuclear research, known as CERN, is a high energy collider specifically designed and constructed to detect the Higgs particle. Two beams of protons circulating at very high speeds in the LHC collide inside two detectors (ATLAS and CMS). Collisions between a proton in one beam and a proton in the other beam result in generation of new particles, possibly including the Higgs boson; each such collision is an event. Some of the particles generated can be tracked and measured in the detectors. However, the Higgs particle, if generated, decays extremely quickly into other known SM particles and cannot be detected directly. Instead, the existence of the Higgs particle must be inferred by looking for those combinations of detectable particles that are predicted by the SM.

Once a Higgs particle has been created (by one of several "production mechanisms") in a proton-proton collision, there are several different processes, called "decay modes", through which the particle may decay. The decay process can be reconstructed based on the detected collision byproducts. Events with reconstructed processes that match one of the possible Higgs decay modes and pass other selection criteria (called cuts) are recorded as "Higgs candidates" and the invariant mass of the unobserved particle is computed from the reconstruction. A histogram of the estimator of the mass is then created for each decay mode. However, there are other processes, not involving the Higgs boson, that can result in the generation of Higgs event byproducts which also pass the cuts; these are called background events. Thus the histogram created is either a mixture of background events and events in which a Higgs particle was created or just a histogram of background events if the Higgs particle does not exist.

In joint work with Richard Lockhart we propose a Bayesian model as well as a Bayesian decision making procedure that incorporates the underlying assumptions provided by the theory and satisfies the strict frequency error rate requirements in particle phisics. The paper can be found here.

Sequentially constrained Monte Carlo (SCMC)

Constraints can be interpreted in a broad sense as any kind of explicit restriction over the parameters. While some constraints are defined directly on the parameter space, when they are instead defined by known behavior on the model, transformation of constraints into features on the parameter space may not be possible. Difficulties in sampling from the posterior distribution as a result of incorporation of constraints into the model is a common challenge leading to truncations in the parameter space and inefficient sampling algorithms. In joint work with Dave Campbell we propose a variant of sequential Monte Carlo algorithm for posterior sampling in presence of constraints by defining a sequence of densities through the imposition of the constraint. Particles generated from an unconstrained or mildly constrained distribution are filtered and moved through sampling and resampling steps to obtain a sample from the fully constrained target distribution. General and model specific forms of constraints enforcing strategies are defined. The SCMC algorithm is demonstrated on constraints defined by monotonicity of a function, densities constrained to low dimensional manifolds, adherence to a theoretically derived model, and model feature matching. For more on SCMC see our paper.

Monotone emulation of computer experiments

In statistical modeling of computer experiments, prior information is sometimes available about the underlying function. For example, the physical system simulated by the computer code may be known to be monotone with respect to some or all inputs. In joint work with Derek Bingham, Hugh Chipman, and Dave Campbell we develop a Bayesian approach to Gaussian process modeling capable of incorporating monotonicity information for computer model emulation. Markov chain Monte Carlo methods are used to sample from the posterior distribution of the process given the simulator output and monotonicity information. The performance of the proposed approach in terms of predictive accuracy and uncertainty quantification is demonstrated in a number of simulated examples as well as a real queuing system application. See our paper here.