Kernel Bayes' rule: Bayesian inference with positive definite kernels

By Kenji Fukumizu, Le Song, Arthur Gretton. 2013

204 citations recorded by the source corpus.

Subfields: Analytical Chemistry, Artificial Intelligence, Probabilistic & Bayesian Methods, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “…(Gretton et al. 2012a), independence and conditional independence tests (Fukumizu et al. 2008, Zhang et al. 2011, Doran et al. 2014), causal inference (Sgouritsa et al. 2013, Chen et al. 2014), adaptive MCMC (Sejdinovic et al. 2014), and approximate Bayesian computation (Fukumizu et al. 2013).”
DISCO Nets : DISsimilarity COefficients Networks (2016, background): “[3], kernel density estimation (KDE) fails in high dimensional output spaces.”
Towards a Neural Statistician (2016, neutral): “See [6] for instance for kernel-based approaches.”
Consistent Kernel Mean Estimation for Functions of Random Variables (2016, methodology): “Although our discussion above focuses on reduced expansion set methods, there are other popular algorithms that produce KME expansions where the samples are not i.i.d.”
Minimax Estimation of Kernel Mean Embeddings (2016, methodology): “…et al., 2008; Fukumizu et al., 2008), covariate-shift (Smola et al., 2007), density estimation (Sriperumbudur, 2011), feature selection (Song et al., 2012), causal inference (Lopez-Paz et al., 2015), kernel Bayes’ rule (Fukumizu et al., 2013) and distribution regression (Szabó et al., 2015).”
Computing functions of random variables via reproducing kernel Hilbert space representations (2015, neutral): “Wecanmapconditionaldistributionsandper-form Bayesian updates (Fukumizu et al. 2008, 2013; Zhang etal.2011),andwecanconnectkernelmapstoFourieroptics, leading to a physical realization as Fraunhofer diffraction (Harmeling et al. 2013).”
Characteristic Kernels and Infinitely Divisible Distributions (2014, methodology): “Note, however, that the kernel mean estimators in applications (Song et al., 2009; Fukumizu et al., 2013) are given by a form of the weighted sum m̂P = ∑n i=1 wik(·, Xi) with data {Xi} n i=1 and associated weights w ∈ Rn. Weights w are allowed to take negative values, and the kernel mean estimator…”
Density Estimation in Infinite Dimensional Exponential Families (2013, methodology): “While such a condition has already been used in various works [8, 32, 19] in the context of non-parametric least squares regression, we explore it in more detail in Proposition 8, and Examples 2 and 3.”
The Nonparametric Kernel Bayes Smoother (2016, methodology): “The same assumption is made in the nKB-filter [10] and the GP-based filter/smoother [17, 6, 7].”
Kernel approximate Bayesian computation in population genetic inferences (2012, methodology): “We have (Fukumizu et al. 2010, 2011) 〈𝑓 ( ∙ ), 𝜇̂ 𝜃 | 𝑠 〉 ℋ − 𝐸 [ 𝑓 ( We want to establish �𝐶̂ �𝐶̂ + 𝜀 First we show �𝐶 �𝐶 + 𝜀 𝐼 � 𝑓 ( ∙ ) − 𝐶 ( 𝐶 + 𝜀 The left hand side is upper bounded by �𝐶̂ �𝐶̂ + 𝜀 𝐼 � �𝐶̂ − 𝐶 � ( 𝐶 𝑠𝑠 + 𝜀 𝐼 𝑛 ) 𝑓 ( ∙ ) � ℋ 𝜃 By the decomposition…”
Kernel Embeddings of Conditional Distributions: A Unified Kernel Framework for Nonparametric Inference in Graphical Models (2013, methodology): “Under the assumption that a training sample is available, cross-validation can be performed on the training sample to select the kernel parameters and regularization parameters for KBF [12].”
Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models (2015, methodology): “…definite, so that the estimated density is a member of a reproducing kernel Hilbert space where more robust approximations of pΦ|θ than the one in Equation (18) would exist (Kim and Scott, 2012), and moreover, one could investigate connections to the inference approach of Fukumizu et al. (2013).”
Smooth Operators (2013, methodology): “We therefore keep the necessary assumptions simple, and we derive rates only for the approximate sum rule, which we compare with the rates of (Fukumizu et al., 2012). We make a number of assumptions, which can be found in Sec. E.1. The main assumption is that HX and HY are finite dimensional. The HY”