40 problems

Let $H$ be a hypergraph with a prescribed degree sequence, and consider the switch Markov chain on the realizations of that degree sequence, whose transitions apply switch operatio…

Let $c=(c_{ijk})$ be the quadratic drift tensor and let … The required sampling task is to draw triples $(i,j,k)$ from the distribution $P$. MCMC sampling conjecture. This sampling…

For each integer $k\geq1$, let $r_k$ denote a real solution of $g(k;r)=g(k+1;r)$, with $r_1=1$, and let $s(r)$ and $g(k;r)$ be as defined in the paper. Threshold sequence conjectur…

Let $r>0$, and let $s(r)$ and $g(k;r)$ be the functions defined in the paper for $k\in\mathbb N$. Exact maximizer conjecture. For all $r\in(0,\infty)$, … The paper notes that the m…

Consider the coordinate-wise Barker proposal, in which each coordinate is separately accepted or reflected according to the current position and gradient information. Reflection-ef…

For the Zig-Zag Sampler (ZZS), let $\lambda$ denote the refresh rate of the velocity. Zero-refresh-rate conjecture. The choice $\lambda=0$ is optimal in terms of minimising the asy…

Let $K$ be a Markov kernel and let the method use a coupling of the corresponding Markov chains, with an expected improvement measured through the effective sample size. In the per…

Let an empirical effective sample size be computed for a system of coupled Markov chains, and consider its degradation under the proposed weight-harmonization method. In practice,…

Let generalized Monte Carlo (GMC) be the discrete-time reflective Hamiltonian Markov chain described in the surrounding construction. An extension to continuous-time dynamics would…

The method substitutes an expensive likelihood function in a Markov chain Monte Carlo algorithm with a neural-operator surrogate. MCMC-Net applicability conjecture. We conjecture t…

Let $q_\theta(w)$ be a proposal density for the multiplicative noise that may depend on the parameter $\theta$, rather than satisfying the assumption that $q_\theta(w)$ is independ…

Let $\mathbb{G}$ be a random $d$-regular graph, and let $\mu_{\mathbb{G}}$ be the ferromagnetic $q$-state Potts distribution with parameter $\beta>0$, where $q$ and $d$ are integer…

Potts sampling conjecture below $\beta_c$. When $\beta<\beta_c$, sampling from the ferromagnetic Potts model admits a polynomial-time algorithm.

Let $\mathcal{V}$ be the unrestricted model space, let an informed Metropolis–Hastings sampler assign proposal probabilities in its neighborhood according to the target posterior p…

Consider partite $3$-regular hypergraphs with vertex classes of sizes $n_1,n_2,n_3$, where the degree in the $i^{\mathrm{th}}$ vertex class is between $c_1 n_{i+1}n_{i+2}$ and…

Let $D$ be a hypergraph degree sequence, and consider the realizations of $D$ together with the switch operations between them. The paper proves irreducibility when $D$ is third al…

Qualitative optimality conjecture. For at least some $d70b_0$ in the class $d70b_{d70b_0,k}$, the result cannot be qualitatively improved.

Consider the MCMC setting for sampling from a distribution associated with a possibly non-concave function $f$, where the available complexity bounds are measured in gradient or fu…

Improved-convergence conjecture. Improved convergence for Markov chain Monte Carlo methods should be observed when

Let MY-MALA denote the Moreau-Yosida Metropolis-adjusted Langevin algorithm, with proximal parameter $\lambda$. A distribution is light-tailed if it belongs to the class of distrib…

Consider the stochastic block model and compare Markov-chain Monte Carlo (MCMC) with message-passing algorithms for inference. MCMC optimality conjecture. MCMC performs as well as…

Convergence-rate conjecture. The convergence rate of MTM-IS($k$) without replacement is faster than

Pseudospectral-gap proxy conjecture. The pseudospectral gap is a good proxy for the convergence rate of Monte Carlo ergodic averages for some proposed nonreversible subsampling MCM…

Let $T$ denote the HMC integration time, subject to the time-step restriction discussed in the analysis. Time-step optimality conjecture. For many problems of interest, this restri…

Higher-order Langevin diffusion conjecture. The exponent on $d$ in the mixing-time complexity can be further reduced, with a trade-off between the dependency on the dimension and t…