### Efficiency and robustness in Monte Carlo sampling of 3-D geophysical inversions with Obsidian v0.1.2: Setting up for success

The rigorous quantification of uncertainty in geophysical inversions is a challenging problem. Inversions are often ill-posed and the likelihood surface may be multi-modal; properties of any single mode become inadequate uncertainty measures, and sampling methods become inefficient for irregular posteriors or high-dimensional parameter spaces. We explore the influences of different choices made by the practitioner on the efficiency and accuracy of Bayesian geophysical inversion methods that rely on Markov chain Monte Carlo sampling to assess uncertainty, using a multi-sensor inversion of the three-dimensional structure and composition of a region in the Cooper Basin of South Australia as a case study. The inversion is performed using an updated version of the Obsidian distributed inversion software. We find that the posterior for this inversion has complex local covariance structure, hindering the efficiency of adaptive sampling methods that adjust the proposal based on the chain history. Within the context of a parallel-tempered Markov chain Monte Carlo scheme for exploring high-dimensional multi-modal posteriors, a preconditioned Crank-Nicholson proposal outperforms more conventional forms of random walk. Aspects of the problem setup, such as priors on petrophysics or on 3-D geological structure, affect the shape and separation of posterior modes, influencing sampling performance as well as the inversion results. Use of uninformative priors on sensor noise can improve inversion results by enabling optimal weighting among multiple sensors even if noise levels are uncertain. Efficiency could be further increased by using posterior gradient information within proposals, which Obsidian does not currently support, but which could be emulated using posterior surrogates.

### Acceleration of expensive computations in Bayesian statistics using vector operations

Many applications in Bayesian statistics are extremely computationally intensive. However, they are also often inherently parallel, making them prime targets for modern massively parallel central processing unit (CPU) architectures. While the use of multi-core and distributed computing is widely applied in the Bayesian community, very little attention has been given to fine-grain parallelisation using single instruction multiple data (SIMD) operations that are available on most modern commodity CPUs. Rather, most fine-grain tuning in the literature has centred around general purpose graphics processing units (GPGPUs). Since the effective utilisation of GPGPUs typically requires specialised programming languages, such technologies are not ideal for the wider Bayesian community. In this work, we practically demonstrate, using standard programming libraries, the utility of the SIMD approach for several topical Bayesian applications. In particular, we consider sampling of the prior predictive distribution for approximate Bayesian computation (ABC), and the computation of Bayesian $p$-values for testing prior weak informativeness. Through minor code alterations, we show that SIMD operations can improve the floating point arithmetic performance resulting in up to $6\times$ improvement in the overall serial algorithm performance. Furthermore $4$-way parallel versions can lead to almost $19\times$ improvement over a na\"{i}ve serial implementation. We illustrate the potential of SIMD operations for accelerating Bayesian computations and provide the reader with essential implementation techniques required to exploit modern massively parallel processing environments using standard software development tools.

### High-dimensional ABC

This Chapter, "High-dimensional ABC", is to appear in the forthcoming Handbook of Approximate Bayesian Computation (2018). It details the main ideas and concepts behind extending ABC methods to higher dimensions, with supporting examples and illustrations.

### Jinbo Huang

National ICT Australia is funded by the Australian Government's Backing Australia '3 Ability initiative, in part through the Australian Research Council. In variable elimination (Zhang & Poole 1996; Dechter 1996), for example, one can use any elimination order to solve the former, but can only choose among orders that put the MAP variables last to solve the latter. Consequently, MPE can be solved in time and space exponential in the treewidth of the network, while the corresponding algorithm for MAP requires time and space exponential in the constrained treewidth, which can be significantly higher. The same gap exists for other stmcture-based methods as well, such as jointree algorithms (Shenoy & Shafer 1986; Jensen, Lauritzen, & Olesen 1990). A recent algorithm proposed in (Park & Darwiche 2003) represents a significant advance of the state of the art in solving MAP exactly.

### Bayesian Models of Data Streams with Hierarchical Power Priors

Making inferences from data streams is a pervasive problem in many modern data analysis applications. But it requires to address the problem of continuous model updating and adapt to changes or drifts in the underlying data generating distribution. In this paper, we approach these problems from a Bayesian perspective covering general conjugate exponential models. Our proposal makes use of non-conjugate hierarchical priors to explicitly model temporal changes of the model parameters. We also derive a novel variational inference scheme which overcomes the use of non-conjugate priors while maintaining the computational efficiency of variational methods over conjugate models. The approach is validated on three real data sets over three latent variable models.