Bayesian inference for doubly-intractable probabilistic graphical models typically involves variations of the exchange algorithm or approximate Markov chain Monte Carlo (MCMC) samplers. However, existing methods for both classes of algorithms require either perfect samplers or sequential samplers for complex models, which are often either not available, or suffer from poor mixing, especially in high dimensions. We develop a method that does not require perfect or sequential sampling, and can be applied to both classes of methods: exact and approximate MCMC. The key to our approach is to utilize the tractable independence model underlying an intractable probabilistic graphical model for the purpose of constructing a finite sample unbiased Monte Carlo (and not MCMC) estimate of the Metropolis--Hastings ratio. This innovation turns out to be crucial for scalability in high dimensions. The method is demonstrated on the Ising model. Gradient-based alternatives to construct a proposal, such as Langevin and Hamiltonian Monte Carlo approaches, also arise as a natural corollary to our general procedure, and are demonstrated as well.

In sampling tasks, it is common for target distributions to be known up to a normalising constant. However, in many situations, evaluating even the unnormalised distribution can be costly or infeasible. This issue arises in scenarios such as sampling from the Bayesian posterior for tall datasets and the 'doubly-intractable' distributions. In this paper, we begin by observing that seemingly different Markov chain Monte Carlo (MCMC) algorithms, such as the exchange algorithm, PoissonMH, and TunaMH, can be unified under a simple common procedure. We then extend this procedure into a novel framework that allows the use of auxiliary variables in both the proposal and acceptance-rejection steps. We develop the theory of the new framework, applying it to existing algorithms to simplify and extend their results. Several new algorithms emerge from this framework, with improved performance demonstrated on both synthetic and real datasets.

Kernel based approximation offers versatile tools for high-dimensional approximation, which can especially be leveraged for surrogate modeling. For this purpose, both "knot insertion" and "knot removal" approaches aim at choosing a suitable subset of the data, in order to obtain a sparse but nevertheless accurate kernel model. In the present work, focussing on kernel based interpolation, we aim at combining these two approaches to further improve the accuracy of kernel models, without increasing the computational complexity of the final kernel model. For this, we introduce a class of kernel exchange algorithms (KEA). The resulting KEA algorithm can be used for finetuning greedy kernel surrogate models, allowing for an reduction of the error up to 86.4% (17.2% on average) in our experiments.

Nowadays, in many different fields, massive data are available and for several reasons, it might be convenient to analyze just a subset of the data. The application of the D-optimality criterion can be helpful to optimally select a subsample of observations. However, it is well known that D-optimal support points lie on the boundary of the design space and if they go hand in hand with extreme response values, they can have a severe influence on the estimated linear model (leverage points with high influence). To overcome this problem, firstly, we propose an unsupervised exchange procedure that enables us to select a nearly D-optimal subset of observations without high leverage values. Then, we provide a supervised version of this exchange procedure, where besides high leverage points also the outliers in the responses (that are not associated to high leverage points) are avoided. This is possible because, unlike other design situations, in subsampling from big datasets the response values may be available. Finally, both the unsupervised and the supervised selection procedures are generalized to I-optimality, with the goal of getting accurate predictions.

Exchange algorithm is one of the most popular extensions of Metropolis-Hastings algorithm to sample from doubly-intractable distributions. However, theoretical exploration of exchange algorithm is very limited. For example, natural questions like `Does exchange algorithm converge at a geometric rate?' or `Does the exchange algorithm admit a Central Limit Theorem?' have not been answered. In this paper, we study the theoretical properties of exchange algorithm, in terms of asymptotic variance and convergence speed. We compare the exchange algorithm with the original Metropolis-Hastings algorithm and provide both necessary and sufficient conditions for geometric ergodicity of the exchange algorithm, which can be applied to various practical applications such as exponential random graph models and Ising models. A central limit theorem for the exchange algorithm is also established. Meanwhile, a concrete example, involving the Binomial model with conjugate and non-conjugate priors, is treated in detail with sharp convergence rates. Our results justify the theoretical usefulness of the exchange algorithm.

Markov chain Monte Carlo (MCMC) algorithms are widely used to sample from complicated distributions, especially to sample from the posterior distribution in Bayesian inference. However, MCMC is not directly applicable when facing the doubly intractable problem. In this paper, we discussed and compared two existing solutions -- Pseudo-marginal Monte Carlo and Exchange Algorithm. This paper also proposes a novel algorithm: Multi-armed Bandit MCMC (MABMC), which chooses between two (or more) randomized acceptance ratios in each step. MABMC could be applied directly to incorporate Pseudo-marginal Monte Carlo and Exchange algorithm, with higher average acceptance probability.

We derive computationally tractable methods to select a small subset of experiment settings from a large pool of given design points. The primary focus is on linear regression models, while the technique extends to generalized linear models and Delta's method (estimating functions of linear regression models) as well. The algorithms are based on a continuous relaxation of an otherwise intractable combinatorial optimization problem, with sampling or greedy procedures as post-processing steps. Formal approximation guarantees are established for both algorithms, and numerical results on both synthetic and real-world data confirm the effectiveness of the proposed methods.

Bayesian inference for models that have an intractable partition function is known as a doubly intractable problem, where standard Monte Carlo methods are not applicable. The past decade has seen the development of auxiliary variable Monte Carlo techniques (M{\o}ller et al., 2006; Murray et al., 2006) for tackling this problem; these approaches being members of the more general class of pseudo-marginal, or exact-approximate, Monte Carlo algorithms (Andrieu and Roberts, 2009), which make use of unbiased estimates of intractable posteriors. Everitt et al. (2017) investigated the use of exact-approximate importance sampling (IS) and sequential Monte Carlo (SMC) in doubly intractable problems, but focussed only on SMC algorithms that used data-point tempering. This paper describes SMC samplers that may use alternative sequences of distributions, and describes ways in which likelihood estimates may be improved adaptively as the algorithm progresses, building on ideas from Moores et al. (2015). This approach is compared with a number of alternative algorithms for doubly intractable problems, including approximate Bayesian computation (ABC), which we show is closely related to the method of M{\o}ller et al. (2006).

A large number of statistical models are "doubly-intractable": the likelihood normalising term, which is a function of the model parameters, is intractable, as well as the marginal likelihood (model evidence). This means that standard inference techniques to sample from the posterior, such as Markov chain Monte Carlo (MCMC), cannot be used. Examples include, but are not confined to, massive Gaussian Markov random fields, autologistic models and Exponential random graph models. A number of approximate schemes based on MCMC techniques, Approximate Bayesian computation (ABC) or analytic approximations to the posterior have been suggested, and these are reviewed here. Exact MCMC schemes, which can be applied to a subset of doubly-intractable distributions, have also been developed and are described in this paper. As yet, no general method exists which can be applied to all classes of models with doubly-intractable posteriors. In addition, taking inspiration from the Physics literature, we study an alternative method based on representing the intractable likelihood as an infinite series. Unbiased estimates of the likelihood can then be obtained by finite time stochastic truncation of the series via Russian Roulette sampling, although the estimates are not necessarily positive. Results from the Quantum Chromodynamics literature are exploited to allow the use of possibly negative estimates in a pseudo-marginal MCMC scheme such that expectations with respect to the posterior distribution are preserved. The methodology is reviewed on well-known examples such as the parameters in Ising models, the posterior for Fisher-Bingham distributions on the $d$-Sphere and a large-scale Gaussian Markov Random Field model describing the Ozone Column data. This leads to a critical assessment of the strengths and weaknesses of the methodology with pointers to ongoing research.