A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines

By G. Kimeldorf, Grace Wahba. 1970

Abstract : The report presents classes of prior distributions for which the Bayes' estimate of an unknown function given certain observations is a spline function. (Author)

Subfields: Statistics and Probability, Probabilistic & Bayesian Methods

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Citations in the corpus

The hardness of conditional independence testing and the generalised covariance measure (2018, background): “d high-dimensional multinomials: Sharp local minimax rates. ArXiv e-prints (1706.10003), 2017. [4]A. R. Barron. Uniformly powerful goodness of t tests. The Annals of Statistics, 17(1): 107{124, 1989. [5]W. P. Bergsma. Testing Conditional Independence for Continuous Random Variables, 2004. EURANDOM-r”
Fastfood: Approximate Kernel Expansions in Loglinear Time (2014, background): “This is because the Representer Theorem of Kimeldorf and Wahba (1970) states that such a function expansion in terms of finitely many coefficients must exist under fairly benign conditions even whenever the space is infinite dimensional.”
Density Estimation in Infinite Dimensional Exponential Families (2013, methodology): “In Theorem 4 (proved in Section 8.5), we provide an alternate expression for f λ,n as a solution of a simple ﬁnite-dimensional linear system (see (7) and (8)), using the ideas in the proof of representer theorem [24, 31].”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, background): “…(Evgeniou et al., 2000), it is well known that kernel-based algorithms (for classiﬁcation/regression) generally invoke the representer theorem (Kimeldorf and Wahba, 1970; Sch¨olkopf et al., 2001) and learn a function in H that has the representation, where N n := { 1 , 2 , . . . , n and { c j…”
Differentially Private Empirical Risk Minimization (2009, neutral): “The representer theorem (Kimeldorf and Wahba, 1970) shows that the regularized empirical risk in (1) is minimized by a function f(x) that is given by a linear combination of kernel functions centered at the data points: f∗(x) = n ∑ i=1 aik(x(i),x).”
Dataset Shift in Machine Learning (2009, methodology): “we can invoke the representer theorem [Kimeldorf and Wahba, 1970] which leads to”
Some Properties of Regularized Kernel Methods (2004, background): “The ﬁrst result in this direction is due to Kimeldorf and Wahba (1970) for the squared loss function (see also Wahba, 1990).”
Non-Gaussian statistical inverse problems. Part I: Posterior distributions (2012, methodology): “In 1957-58 Whittle [136, 137] discussed prior information on the smoothness of the unknown in certain density estimation problems, and later Kimeldorf and Wahba [71] clarified the relations between smoothing and Gaussian prior models.”
Efficient algorithms for ranking with SVMs (2010, background): “The key observation comes from the representer theorem (Kimeldorf and Wahba, 1970): since the objective function (1) only depends on the regularizer and the function evaluation at the training samples x i , this theorem states that the function minimizing (1) in H can be written as: f (x) = n X i=1…”
On the Similarity between the Laplace and Neural Tangent Kernels (2020, background): “, the Gaussian and Laplace kernels, are typically derived for input in R [29].”
Diagnostics for Gaussian Process Emulators (2009, background): “From a Bayesian point of view Kimeldorf and Wahba (1970) and O’Hagan (1978) use Gaussian processes to describe the behaviour of an unknown mathematical function.”
Fully Bayesian spline smoothing and intrinsic autoregressive priors (2003, background): “(Kimeldorf & Wahba, 1970; Wahba, 1978) is to take a Gaussian process prior on g(.)”
Whole-Genome Regression and Prediction Methods Applied to Plant and Animal Breeding (2013, background): “Most penalized estimates are equivalent to posterior modes of certain types of Bayesian models (Kimeldorf and Wahba 1970; Tibshirani 1996).”