The stochastic block model (SBM) is widely used for modelling network data by assigning individuals (nodes) to communities (blocks) with the probability of an edge existing between individuals depending upon community membership. In this paper we introduce an autoregressive extension of the SBM, based on continuous-time Markovian edge dynamics. The model is appropriate for networks evolving over time and allows for edges to turn on and off. Moreover, we allow for the movement of individuals between communities. An effective reversible jump Markov chain Monte Carlo algorithm is introduced for sampling jointly from the posterior distribution of the community parameters and the number and location of changes in community membership. The algorithm is successfully applied to a network of mice.

We introduced the Hug and Hop Markov chain Monte Carlo algorithm for estimating expectations with respect to an intractable distribution π. The algorithm alternates between two kernels: Hug and Hop. Hug is a non-reversible kernel that uses repeated applications of the bounce mechanism from the recently proposed Bouncy Particle Sampler to produce a proposal point far from the current position, yet on almost the same contour of the target density, leading to a high acceptance probability. Hug is complemented by Hop, which deliberately proposes jumps between contours and has an efficiency that degrades very slowly with increasing dimension. There are many parallels between Hug and Hamiltonian Monte Carlo (HMC) using a leapfrog intergator, including an O(δ 2 ) error in the integration scheme, however Hug is also able to make use of local Hessian information without requiring implicit numerical integration steps, improving efficiency when the gains in mixing outweigh the additional computational costs. We test Hug and Hop empirically on a variety of toy targets and real statistical models and find that it can, and often does, outperform HMC on the exploration of components of the target.

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Amongst Markov chain Monte Carlo algorithms, Hamiltonian Monte Carlo (HMC) is often the algorithm of choice for complex, high-dimensional target distributions; however, its efficiency is notoriously sensitive to the choice of the integration-time tuning parameter, T . When integrating both forward and backward in time using the same leapfrog integration step as HMC, the set of local maxima in the potential along a path, or apogees, is the same whatever point (position and momentum) along the path is chosen to initialise the integration. We present the Apogee to Apogee Path Sampler (AAPS), which utilises this invariance to create a simple yet generic methodology for constructing a path, proposing a point from it and accepting or rejecting that proposal so as to target the intended distribution. We demonstrate empirically that AAPS has a similar efficiency to HMC but is much more robust to the setting of its equivalent tuning parameter, a non-negative integer, K, the number of apogees that the path crosses.

Summary We introduced the Hug and Hop Markov chain Monte Carlo algorithm for estimating expectations with respect to an intractable distribution. The algorithm alternates between two kernels: Hug and Hop. Hug is a non-reversible kernel that repeatedly applies the bounce mechanism from the recently proposed Bouncy Particle Sampler to produce a proposal point far from the current position, yet on almost the same contour of the target density, leading to a high acceptance probability. Hug is complemented by Hop, which deliberately proposes jumps between contours and has an efficiency that degrades very slowly with increasing dimension. There are many parallels between Hug and Hamiltonian Monte Carlo using a leapfrog integrator, including the order of the integration scheme, however Hug is also able to make use of local Hessian information without requiring implicit numerical integration steps, and its performance is not terminally affected by unbounded gradients of the log-posterior. We test Hug and Hop empirically on a variety of toy targets and real statistical models and find that it can, and often does, outperform Hamiltonian Monte Carlo.