Illustration of the two transition kernels PMALA and MTM on a two-dimensional probability distribution: a Gaussian mixture model (shown with the red ellipses) restricted to validity intervals (shown with the large black square). Top: MTM update rule (on the right), and a histogram after 100 sampling steps (on the left). With this update rule only, the sampling visits all the distribution modes at the cost of computationally heavier individual steps. Middle: PMALA update rule (on the right), and a histogram after 100 sampling steps (on the left). With this cheaper update rule only, the sampling fails to visit all the distribution modes. Bottom: Two histograms obtained using both kernels after 100 steps (left) and 10 000 steps (right). The histograms converge to the correct distribution.