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Register a new userA Bayesian Nonparametric Estimation of Mutual Information
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Mutual information is a widely-used information theoretic measure to quantify the amount of association between variables. It is used extensively in many applications such as image registration, diagnosis of failures in electrical machines, pattern recognition, data mining and tests of independence. The main goal of this paper is to provide an efficient estimator of the mutual information based on the approach of Al Labadi et. al. (2021). The estimator is explored through various examples and is compared to its frequentist counterpart due to Berrett et al. (2019). The results show the good performance of the procedure by having a smaller mean squared error.
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Mutual information is a well-known tool to measure the mutual dependence between variables. In this paper, a Bayesian nonparametric estimation of mutual information is established by means of the Dirichlet process and the $k$-nearest neighbor distance. As a direct outcome of the estimation, an easy-to-implement test of independence is introduced through the relative belief ratio. Several theoretical properties of the approach are presented. The procedure is investigated through various examples where the results are compared to its frequentist counterpart and demonstrate a good performance.
Conditional Mutual Information (CMI) is a measure of conditional dependence between random variables X and Y, given another random variable Z. It can be used to quantify conditional dependence among variables in many data-driven inference problems such as graphical models, causal learning, feature selection and time-series analysis. While k-nearest neighbor (kNN) based estimators as well as kernel-based methods have been widely used for CMI estimation, they suffer severely from the curse of dimensionality. In this paper, we leverage advances in classifiers and generative models to design methods for CMI estimation. Specifically, we introduce an estimator for KL-Divergence based on the likelihood ratio by training a classifier to distinguish the observed joint distribution from the product distribution. We then show how to construct several CMI estimators using this basic divergence estimator by drawing ideas from conditional generative models. We demonstrate that the estimates from our proposed approaches do not degrade in performance with increasing dimension and obtain significant improvement over the widely used KSG estimator. Finally, as an application of accurate CMI estimation, we use our best estimator for conditional independence testing and achieve superior performance than the state-of-the-art tester on both simulated and real data-sets.
Estimation of information theoretic quantities such as mutual information and its conditional variant has drawn interest in recent times owing to their multifaceted applications. Newly proposed neural estimators for these quantities have overcome severe drawbacks of classical $k$NN-based estimators in high dimensions. In this work, we focus on conditional mutual information (CMI) estimation by utilizing its formulation as a minmax optimization problem. Such a formulation leads to a joint training procedure similar to that of generative adversarial networks. We find that our proposed estimator provides better estimates than the existing approaches on a variety of simulated data sets comprising linear and non-linear relations between variables. As an application of CMI estimation, we deploy our estimator for conditional independence (CI) testing on real data and obtain better results than state-of-the-art CI testers.
We present a new Bayesian nonparametric approach to estimating the spectral density of a stationary time series. A nonparametric prior based on a mixture of B-spline distributions is specified and can be regarded as a generalization of the Bernstein polynomial prior of Petrone (1999a,b) and Choudhuri et al. (2004). Whittles likelihood approximation is used to obtain the pseudo-posterior distribution. This method allows for a data-driven choice of the number of mixture components and the location of knots. Posterior samples are obtained using a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm, and mixing is improved using parallel tempering. We conduct a simulation study to demonstrate that for complicated spectral densities, the B-spline prior provides more accurate Monte Carlo estimates in terms of $L_1$-error and uniform coverage probabilities than the Bernstein polynomial prior. We apply the algorithm to annual mean sunspot data to estimate the solar cycle. Finally, we demonstrate the algorithms ability to estimate a spectral density with sharp features, using real gravitational wave detector data from LIGOs sixth science run, recoloured to match the Advanced LIGO target sensitivity.
We point out a limitation of the mutual information neural estimation (MINE) where the network fails to learn at the initial training phase, leading to slow convergence in the number of training iterations. To solve this problem, we propose a faster method called the mutual information neural entropic estimation (MI-NEE). Our solution first generalizes MINE to estimate the entropy using a custom reference distribution. The entropy estimate can then be used to estimate the mutual information. We argue that the seemingly redundant intermediate step of entropy estimation allows one to improve the convergence by an appropriate reference distribution. In particular, we show that MI-NEE reduces to MINE in the special case when the reference distribution is the product of marginal distributions, but faster convergence is possible by choosing the uniform distribution as the reference distribution instead. Compared to the product of marginals, the uniform distribution introduces more samples in low-density regions and fewer samples in high-density regions, which appear to lead to an overall larger gradient for faster convergence.
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