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The density matrix in quantum mechanics parameterizes the statistical properties of the system under observation, just like a classical probability distribution does for classical systems. The expectation value of observables cannot be measured directly, it can only be approximated by applying classical statistical methods to the frequencies by which certain measurement outcomes (clicks) are obtained. In this paper, we make a detailed study of the statistical fluctuations obtained during an experiment in which a hypothesis is tested, i.e. the hypothesis that a certain setup produces a given quantum state. Although the classical and quantum problem are very much related to each other, the quantum problem is much richer due to the additional optimization over the measurement basis. Just as in the case of classical hypothesis testing, the confidence in quantum hypothesis testing scales exponentially in the number of copies. In this paper, we will argue 1) that the physically relevant data of quantum experiments is only contained in the frequencies of the measurement outcomes, and that the statistical fluctuations of the experiment are essential, so that the correct formulation of the conclusions of a quantum experiment should be given in terms of hypothesis tests, 2) that the (classical) $chi^2$ test for distinguishing two quantum states gives rise to the quantum $chi^2$ divergence when optimized over the measurement basis, 3) present a max-min characterization for the optimal measurement basis for quantum goodness of fit testing, find the quantum measurement which leads both to the maximal Pitman and Bahadur efficiency, and determine the associated divergence rates.
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The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness-of-fit test for the finite-lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness-of-fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness-of-fit testing for the Ising model which avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.
We investigate the statistics of stationary points in the sum of squares of $N$ Gaussian random fields, which we call a chi-squared field. The behavior of such a field at a point is investigated, with particular attention paid to the formation of topological defects. An integral to compute the number density of stationary points at a given field amplitude is constructed. We compute exact expressions for the integral in various limits and provide code to evaluate it numerically in the general case. We investigate the dependence of the number density of stationary points on the field amplitude, number of fields, and power spectrum of the individual Gaussian random fields. This work parallels the work of Bardeen, Bond, Kaiser and Szalay, who investigated the statistics of peaks of Gaussian random fields. A number of results for integrating over matrices are presented in appendices.
We propose and study a general method for construction of consistent statistical tests on the basis of possibly indirect, corrupted, or partially available observations. The class of tests devised in the paper contains Neymans smooth tests, data-driven score tests, and some types of multi-sample tests as basic examples. Our tests are data-driven and are additionally incorporated with model selection rules. The method allows to use a wide class of model selection rules that are based on the penalization idea. In particular, many of the optimal penalties, derived in statistical literature, can be used in our tests. We establish the behavior of model selection rules and data-driven tests under both the null hypothesis and the alternative hypothesis, derive an explicit detectability rule for alternative hypotheses, and prove a master consistency theorem for the tests from the class. The paper shows that the tests are applicable to a wide range of problems, including hypothesis testing in statistical inverse problems, multi-sample problems, and nonparametric hypothesis testing.
Reduced chi-squared is a very popular method for model assessment, model comparison, convergence diagnostic, and error estimation in astronomy. In this manuscript, we discuss the pitfalls involved in using reduced chi-squared. There are two independent problems: (a) The number of degrees of freedom can only be estimated for linear models. Concerning nonlinear models, the number of degrees of freedom is unknown, i.e., it is not possible to compute the value of reduced chi-squared. (b) Due to random noise in the data, also the value of reduced chi-squared itself is subject to noise, i.e., the value is uncertain. This uncertainty impairs the usefulness of reduced chi-squared for differentiating between models or assessing convergence of a minimisation procedure. The impact of noise on the value of reduced chi-squared is surprisingly large, in particular for small data sets, which are very common in astrophysical problems. We conclude that reduced chi-squared can only be used with due caution for linear models, whereas it must not be used for nonlinear models at all. Finally, we recommend more sophisticated and reliable methods, which are also applicable to nonlinear models.
Non-parametric goodness-of-fit testing procedures based on kernel Stein discrepancies (KSD) are promising approaches to validate general unnormalised distributions in various scenarios. Existing works have focused on studying optimal kernel choices to boost test performances. However, the Stein operators are generally non-unique, while different choices of Stein operators can also have considerable effect on the test performances. In this work, we propose a unifying framework, the generalised kernel Stein discrepancy (GKSD), to theoretically compare and interpret different Stein operators in performing the KSD-based goodness-of-fit tests. We derive explicitly that how the proposed GKSD framework generalises existing Stein operators and their corresponding tests. In addition, we show thatGKSD framework can be used as a guide to develop kernel-based non-parametric goodness-of-fit tests for complex new data scenarios, e.g. truncated distributions or compositional data. Experimental results demonstrate that the proposed tests control type-I error well and achieve higher test power than existing approaches, including the test based on maximum-mean-discrepancy (MMD).
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