A generative model for 3D urban scene understanding from movable platforms Supplementary Material

Bayesian inference often requires the calculation of posterior distributions. Unfortunately, only in special cases (e. g., conjugate priors) it is possible to compute analytically the posterior. Typical approximations to this computation are Laplace approximations and sampling. In sampling-based approaches the integral is approximated numerically. Markov chain Monte Carlo methods (MCMC) can be used to generate samples from a complex graphical model efficiently. The basic idea is to create an artificial ergodic Markov chain so that the posterior distribution of interest becomes a stationary distribution of the Markov chain. Hence, after a burn-in period sampling from the Markov chain yields samples from the posterior distribution [1]. The main challenge in MCMC methods is to create an adequate transition kernel for the Markov chain. Two major strategies have been developed, Metropolis-Hastings and Gibbs sampling. Metropolis-Hastings [4] uses an arbitrary proposal distribution q(x′|x) to sample a new candidate state x′ from the current state x of the Markov chain. x′ is accepted as the new state of the Markov chain with a certain probability A(x, x′), otherwise the current state x is replicated. The acceptance probabilityA(x, x′) is designed in such a way that the stochastic transition kernel T (·|x) defined by this procedure meets the detailed balance condition for Markov chains, i. e.,