45 problems

Consider model-based clustering by maximum-likelihood estimation in mixture models, with $D_m$ the number of parameters in model $m$, $p_2(m)$ the corresponding empirical penalty q…

The Student family of probability laws is infinitely divisible but is not closed under convolution. Consider instead families consisting of mixtures of Student distributions. Closu…

Let $p(x,t\mid\nu_0)$ be the marginal density of a Lévy–Student process at time $t$, represented as a mixture of Student densities $f(x\mid\nu,\delta)$ with mixing distribution…

A Lévy–Student process is the process whose one-dimensional marginal density at time $t$ is denoted by $p(x,t\mid\nu_0)$, with $\nu_0$ the initial Student index. Let…

The data consist of $n=2809$ genes measured under $d=19$ conditions, with an interaction model describing gene profiles across conditions. Biologists' conjecture. If the interactio…

Let the weights of a stick-breaking prior be given by … including, in particular, the Pitman--Yor choice $a_k=1-\sigma$ and $b_k=\alpha+\sigma k$ with $\sigma\in[0,1)$. Suppose the…

Minimax-rate conjecture. This sub-Exponential rate of convergence should be minimax optimal up to constants that may depend on the parameters of the mixture model.

Finite Gaussian mixtures with unknown locations and unknown covariance matrices form the location-scale setting. In this setting, covariance perturbations create higher-order inter…

Consider Gaussian and Poisson mixture models with a bounded parameter space $\Theta$, and let the NPMLE denote the nonparametric maximum likelihood estimator of the mixing distribu…

Let $g_0$ be the true mixing distribution in a Gaussian or Poisson mixture model, and suppose that $g_0$ is finitely discrete. Let $\hat g$ denote the nonparametric maximum likelih…

Consider the nonparametric density demixing model with $K$ component densities, sample-size parameters $N$ and $n$, and a plug-in estimator of the mixing matrix $\Pi$. Plug-in erro…

Let the conditional distribution of the observed responses in a dependent mixture model belong to an exponential family, and suppose that its partition function is nontrivial and e…

Let the MLE be the maximum likelihood estimator for multivariate mixtures of power series distributions with a conditional independence structure, and let $\ell_p$ denote the relev…

Finite-component conjecture. For every $\alpha\in[0,1/2]$, the estimator $\widetilde{\pi}_{n,\alpha}$ has no more than $N=X_{(n)}+1$ components.

Beta-prior conjecture. The appropriate prior for mixtures of Geometric and Negative Binomial distributions is most likely a $\text{Beta}(a,b)$ prior, with $a,b otin (0,infty)$.

Ye et al.'s rate conjecture. A better rate than $\mathcal{O}(h+1/\sqrt{t}+1/ht)$ should not be attainable even when the mixing distribution has higher-order smoothness.

A subgaussian distribution is a distribution whose one-dimensional marginals have uniformly subgaussian tails, and a mixture-clustering regime is specified by the separation betwee…

Let $k$ be the number of components in a mixture, and let $d$ denote the number of moments used. Rational identifiability means that, for generic sample moments for which the momen…

Consider $k$-mixtures of the Gaussian, gamma, and inverse Gaussian distributions, and let $d$ denote the number of moments used. Rational identifiability means that, for generic sa…

For $k$-mixtures of the inverse Gaussian distribution or the gamma distribution, consider the moments through order $3k$, and let rational identifiability mean that a general momen…

Let $\mathcal{M}_{n,d}$ be the mixture variety, let $\sigma_r(\mathcal{M}_{n,d})$ denote its $r$th secant variety, and let the integer linear program in the source provide a bound…

Let $I$ be the ideal of the mixture variety $\mathcal{M}_{n,d}$, and consider its generators as $n$ and $d$ vary. Uniform degree-bound conjecture. There does not exist a uniform de…

The problem concerns learning mixtures of linear regressions in the non-realizable setting, with objective given by the empirical risk minimization formulation discussed above. The…

Worst-case exponent conjecture. At the worst case, $c$ should be strictly smaller than $1/2$. The paper gives explicit attainable exponents for Poisson and negative-binomial mixtur…

Let a stochastic mixture have component family $\mathrm{q}(\mathbf{x};\theta)$ and mixing distribution $\psi(\theta)$. Call it unbiased when its mixture density satisfies…