GNGTS 2019 - Atti del 38° Convegno Nazionale

GNGTS 2019 S essione 3.3 747 In the second analytical example we consider a 2D multivariate Gaussian-mixture distribution in which the two modes are separated by a low-probability region. In this case all the algorithms use 10 chains running for 30000 iterations each, whereas the burn-in period is fixed to 15000 iterations. From Fig. 2a we observe that RWM, AM_sd, and AM do not accurately sample the target pdf and provide biased estimations. More in detail, all the three methods successfully sample the target marginal pdf along the y axis where a single mode occurs. Differently, these algorithms are not capable of sampling the target distribution along the x axis where the marginal pdf shows two modes separated by a high energy barrier. In other terms, the RWM, AM, and AM_sd algorithms get trapped within a specific mode, while the low probability region forms a barrier that prevents the algorithm to explore the other peak. As a result, one of the two modes is oversampled and this results in a overestimation of the associated actual probability value. Differently, both DEMC and DREAM provide final estimated pdfs very close to the target probability density. This is a further demonstration that the mixing of the information brought by the different chains is a successful strategy that guarantees an accurate sampling of complex target pdf . The analysis of the evolution of the PSRF values (Fig. 2b) again proves that all the algorithms accurately sample the target pdf along the y direction, while only the DREAM and DEMC are capable of converging toward the desired probability density along the x axis. Finally, because of the marginal target pdf along the y axis is very simple (a unimodal Gaussian) all the five methods reach the desired PSRF value after 700 iterations (after the burn-in period). Fig. 1 - Results for the 1D analytical distribution: a) Comparison of sampled and target pdfs . B) Evolution of the PSRF value after the burn-in period.