On Bayesian Inference for Continuous-Time Autoregressive Models without Likelihood

Continuous-time autoregressive (CAR) model is very powerful when modeling many real world continuous processes. When the model is driven by Brownian motion, parameter inference is usually based on the likelihood calculation using the Kalman filter; while the model is driven by non-Gaussian Lévy process, Monte Carlo type of methods are often applied to approximate the likelihood. In both cases, likelihood evaluation is the key but is not always easy. Here we propose an innovative Bayesian inference method without the requirement of likelihood evaluation. The algorithm is in a framework of approximate Bayesian computation (ABC). Distance correlation is employed as a very flexible summary statistics for ABC and the p-value calculated from distance correlation provides a good measurement of the dependence between generated samples. Simulation study shows that this approach is straightforward and effective in inferring CAR model parameters.