Sequential methods aim to update beliefs about a set of parameters given new blocks of data that arise in sequence. Early research in this area was motivated by the case where the blocks of data arise in time and as a result of observing an underlying dynamical system, but an important modern application is in the analysis of large datasets. This thesis considers both the design and application of sequential methods.

A new adaptive sequential Monte Carlo (SMC) methodology is presented. By incorporating adaptive Markov chain Monte Carlo (MCMC) moves into the SMC update, it is possible to utilise the heuristic, computational and theoretical advantages of SMC to make gains in sampling efficiency. The new method is tested on the problem of Bayesian mixture analysis and found to outperform an adaptive MCMC algorithm in 5 out of 6 of the situations considered. Theoretical justification of the method, guidelines for implementation and a condition for convergence are provided.

When the dimensionality of the parameter space is high, methods such as the adaptive SMC sampler do not work well. In such cases, sequential data analysis can proceed with statistical models that are amenable to the exact or approximate filtering recursions. The two situations considered here will be the rating of sports teams and players.

A new method for rating and selecting teams for the NCAA basketball tournament is considered. The selection of teams is important to University institutions in the United States, as admittance brings academic as well as sports-related financial benefits. Currently the selection process is undertaken by a panel of expert voters. The new method is in the main found to agree with these pundits, but in the seasons considered a small number of cases are highlighted where injustice to the team was evident.

Also considered is the rating of professional basketball players. A new method is developed that measures a player's offensive and defensive ability and provides a means of combining this information into an overall rating. The method uses data from multiple seasons to more accurately estimate player abilities in a single season. Injustice in the assigning of NBA awards in the 2009 season is uncovered, but the research also highlights one possible reason for this: the commonly cited box-score statistics contain little information on defensive ability.