Related papers: Computing the Dirichlet-Multinomial Log-Likelihood…
In the idealized Morgan model of crossover, we study the probability distributions of shared DNA (identical by descent) between individuals having a wide range of relationships (not just lineal descendants), especially cases for which…
The use of dual system estimation (DSE) is heavily used in Census Bureau operations. With DSE methods, it is important to implement methods to infer the population size among those with missing data from one or both data sources. The use of…
This paper concerns the mathematical analyses of the diffusion model in machine learning. The drift term of the backward sampling process is represented as a conditional expectation involving the data distribution and the forward diffusion.…
Network diffusion models are applicable to many socioeconomic interactions, yet network interaction is hard to observe or measure. Whenever the diffusion process is unobserved, the number of possible realizations of the latent matrix that…
Equipping the probability space with a local Dirichlet form with square field operator $\Gamma$ and generator $A$ allows to improve Monte Carlo computations of expectations, densities, and conditional expectations, as soon as we are able to…
Large Language Models (LLMs) are a powerful tool for statistical text analysis, with derived sequences of next-token probability distributions offering a wealth of information. Extracting this signal typically relies on metrics such as…
We propose a framework for computing, optimizing and integrating with respect to a smooth marginal likelihood in statistical models that involve high-dimensional parameters/latent variables and continuous low-dimensional hyperparameters.…
The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions-i.e. functions that are shaped like the gamma…
In this paper, we discuss the construction of a multivariate generalisation of the Dirichlet-multinomial distribution. An example from forensic genetics in the statistical analysis of DNA mixtures motivates the study of this multivariate…
Denoising diffusion probabilistic models (DDPM) are a class of generative models which have recently been shown to produce excellent samples. We show that with a few simple modifications, DDPMs can also achieve competitive log-likelihoods…
We derive the conjugate prior of the Dirichlet and beta distributions and explore it with numerical examples to gain an intuitive understanding of the distribution itself, its hyperparameters, and conditions concerning its convergence. Due…
The normalized maximum likelihood (NML) is one of the most important distribution in coding theory and statistics. NML is the unique solution (if exists) to the pointwise minimax regret problem. However, NML is not defined even for simple…
Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood…
Dirichlet distributions are probability measures on the unit simplex. They are often used as prior distributions in modeling categorical data, such as in topic analysis of text data. Motivated by this application, we consider Monte Carlo…
We present a general modified maximum likelihood (MML) method for inferring generative distribution functions from uncertain and biased data. The MML estimator is identical to, but easier and many orders of magnitude faster to compute than…
In general, while obtaining the probability density function of sums and products of shifted random variables, ordinary analytical methods such as Fourier and Mellin transforms tend to provide integrals which cannot be expressed in terms of…
This study presents new closed-form estimators for the Dirichlet and the Multivariate Gamma distribution families, whose maximum likelihood estimator cannot be explicitly derived. The methodology builds upon the score-adjusted estimators…
With the widespread success of deep neural networks in science and technology, it is becoming increasingly important to quantify the uncertainty of the predictions produced by deep learning. In this paper, we introduce a new method that…
This paper develops the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially…
We present a mixed multinomial logit (MNL) model, which leverages the truncated stick-breaking process representation of the Dirichlet process as a flexible nonparametric mixing distribution. The proposed model is a Dirichlet process…