Recently, a great deal of attention has been paid to the stage-sequential process for the longitudinal data and a number of methods for analyzing stage-sequential processes have been derived from the family of finite mixture modeling. However, research on the sequential process is rendered difficult by the fact that the number of latent components is not knowna priori. To address this problem, we propose two solutions, reversible jump MCMC and the Bayesian non-parametric approach, so as to provide a set of principles for the systematic model selection for the stage-sequential process. The reversible jump MCMC sampler can explore parameter space and automatically learn the model. Nevertheless, we have found that reversible jump Markov chain Monte Carlo requires the efficient design of proposalmechanism as jumping rules. To reduce the technical and computational burdens, we propose a Bayesian non-parametric approach to select the number of latent components. Using a latent class-profile analysis, we test both algorithms on synthesized data sets to evaluate their performances in model selection problems.Once a model is selected, the model parameters are needed to be estimated. The expectation-maximization algorithm (Dempster et al., 1977) and the data augmentation using MCMC (Hastings, 1970; Tanner and Wong, 1987a) are widely-used techniques to draw statistical inferences of the parameters for the LCPA model. As a number of measurement occasions increases in the LCPA model, however, the computation cost of expectation-maximization or MCMC will become exponentially intensive. On the contrary, if one adapts recursive scheme in the update steps, calculations will be simplified and become generalized to more time points. In light of this, we formulate each update step with recursive terms which are directly analogous to forward-backward algorithm (Chib, 1996; MacKay, 1997).The parameter estimation for the LCPA model benefits from recursive formula, but the recursive algorithm still requires careful examination for the existence of multiple local modes of the objective function (i.e., log-likelihood). Applying the recursive formula, we implement deterministic annealing EM (Ueda and Nakano, 1998) and deterministic annealing variant of variational Bayes (Katahiral et al., 2008) in order to find parameter estimates on the global mode of the objective function. Both methods are based on the deterministic annealing framework, in which ω is included as an annealing parameter to control the annealing rate. By adjusting the value of ω, the annealing process tracks multiple local modes and identifies the globalized optimum as a result.At last, we are interested in analyzing the early onset drinking behaviours among the young generation. We apply latent class-profile analysis to alcohol drinking behaviours as manifest in self-reported items drawn from the National Longitudinal Survey of Youth 1997, which was a survey that explores the transition from school to work and from adolescence to adulthood in the USA. To unveil the stage-sequential bevaviroal progressions, we adopt dynamic Dirichlet learning process to characterize the probable progressions in a discrete manner and then identify patterns in which similar progressions are grouped. For the parameter estimations, we conduct deterministic annealing approaches with predetermined annealing schedule.
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