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In this study, Bayesian estimation is performed for a class of random coefficient self-exciting integer-valued threshold autoregressive processes with explanatory variables. A new model with a linear structure is obtained through model reconstruction, which makes Markov Chain Monte Carlo method easy to perform. By introducing the latent variables series, a complete data likelihood is obtained. Based on this likelihood, the full conditional distributions are easily obtained for all the parameters and latent variables. By maximizing the posterior probability function, the threshold parameter is accurately estimated. Finally, some numerical results of the estimates and a real data example of crime counts in Ballina, New South Wales, Australia are presented. [ABSTRACT FROM AUTHOR]