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Register a new userPhilosophy and the practice of Bayesian statistics
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Added by
Cosma Rohilla Shalizi
Publication date
2010
fields
Physics
and research's language is
English
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A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico-deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science. Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.
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In this paper, we study Kaplan-Meier V- and U-statistics respectively defined as $theta(widehat{F}_n)=sum_{i,j}K(X_{[i:n]},X_{[j:n]})W_iW_j$ and $theta_U(widehat{F}_n)=sum_{i eq j}K(X_{[i:n]},X_{[j:n]})W_iW_j/sum_{i eq j}W_iW_j$, where $widehat{F}_n$ is the Kaplan-Meier estimator, ${W_1,ldots,W_n}$ are the Kaplan-Meier weights and $K:(0,infty)^2tomathbb R$ is a symmetric kernel. As in the canonical setting of uncensored data, we differentiate between two asymptotic behaviours for $theta(widehat{F}_n)$ and $theta_U(widehat{F}_n)$. Additionally, we derive an asymptotic canonical V-statistic representation of the Kaplan-Meier V- and U-statistics. By using this representation we study properties of the asymptotic distribution. Applications to hypothesis testing are given.
The choice of the summary statistics used in Bayesian inference and in particular in ABC algorithms has bearings on the validation of the resulting inference. Those statistics are nonetheless customarily used in ABC algorithms without consistency checks. We derive necessary and sufficient conditions on summary statistics for the corresponding Bayes factor to be convergent, namely to asymptotically select the true model. Those conditions, which amount to the expectations of the summary statistics to asymptotically differ under both models, are quite natural and can be exploited in ABC settings to infer whether or not a choice of summary statistics is appropriate, via a Monte Carlo validation.
Velocity measurements of wind blowing near the North Sea border of Northern Germany and velocity measurements under local isotropic conditions of a turbulent wake behind a cylinder are compared. It is shown that wind gusts - measured by means of velocity increments - do show similar statistics to the laboratory data, if they are conditioned on an averaged wind speed value. Clear differences between the laboratory data and the atmospheric wind velocity measurement are found for the waiting time statistics between successive gusts above a certain threshold of interest.
Random divisions of an interval arise in various context, including statistics, physics, and geometric analysis. For testing the uniformity of a random partition of the unit interval $[0,1]$ into $k$ disjoint subintervals of size $(S_k[1],ldots,S_k[k])$, Greenwood (1946) suggested using the squared $ell_2$-norm of this size vector as a test statistic, prompting a number of subsequent studies. Despite much progress on understanding its power and asymptotic properties, attempts to find its exact distribution have succeeded so far for only small values of $k$. Here, we develop an efficient method to compute the distribution of the Greenwood statistic and more general spacing-statistics for an arbitrary value of $k$. Specifically, we consider random divisions of ${1,2,dots,n}$ into $k$ subsets of consecutive integers and study $|S_{n,k}|^p_{p,w}$, the $p$th power of the weighted $ell_p$-norm of the subset size vector $S_{n,k}=(S_{n,k}[1],ldots,S_{n,k}[k])$ for arbitrary weights $w=(w_1,ldots,w_k)$. We present an exact and quickly computable formula for its moments, as well as a simple algorithm to accurately reconstruct a probability distribution using the moment sequence. We also study various scaling limits, one of which corresponds to the Greenwood statistic in the case of $p=2$ and $w=(1,ldots,1)$, and this connection allows us to obtain information about regularity, monotonicity and local behavior of its distribution. Lastly, we devise a new family of non-parametric tests using $|S_{n,k}|^p_{p,w}$ and demonstrate that they exhibit substantially improved power for a large class of alternatives, compared to existing popular methods such as the Kolmogorov-Smirnov, Cramer-von Mises, and Mann-Whitney/Wilcoxon rank-sum tests.
Two-sample tests utilizing a similarity graph on observations are useful for high-dimensional data and non-Euclidean data due to their flexibility and good performance under a wide range of alternatives. Existing works mainly focused on sparse graphs, such as graphs with the number of edges in the order of the number of observations. However, the tests have better performance with denser graphs under many settings. In this work, we establish the theoretical ground for graph-based tests with graphs that are much denser than those in existing works.
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