%0 Book Section %1 statphys23_0404 %A Aida, T. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K bayesian criteria field group inference information models non-parametric renormalization statphys23 theory topic-11 %T Renormalization and Renormalization Group in Non-parametric Bayesian Statistical Inference %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=404 %X When we observe a set of data points, we usually adopt a model with a finite number of parameters and fit the parameters optimally so as to explain the data. Here, it is known that the best performance is achieved for an optimal number of the parameters: The more complicated data are observed, the more parameters are needed, and vice versa. The value of the optimal number is predicted by `information criteria' such as Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), and so on. On the other hand, the best performance is achieved by parameter tuning, when the data are produced from the same model as we hypothesized. We can tune the parameters of the model, according to the prescription such as the maximization of marginal likelihood. Both optimizations above appear to be different from each other. But they can be understood from a single point of view of `parameter scaling', which was shown in a pioneering work by Nemenman and Bialek 1 for density estimation problem. In this presentation, we show that the above two kinds of optimization are described in a unified form by a new type of information criteria related to the renormalization group equation. In the context of non-parametric models (e.g. 2), we make clear the notions and roles of renormalization and renormalization group in Bayesian statistical inference, which lead to the improvement of learning performance of a broader class of models. References:\\ 1) I. Nemenman and W. Bialek, Phys.Rev.E. 65 (2002) 026137.\\ 2) W. Bialek, C.G. Callan and S.P. Strong, Phys.Rev.Lett. 77 (1996) 4693.

@incollection{statphys23_0404, abstract = {When we observe a set of data points, we usually adopt a model with a finite number of parameters and fit the parameters optimally so as to explain the data. Here, it is known that the best performance is achieved for an optimal number of the parameters: The more complicated data are observed, the more parameters are needed, and vice versa. The value of the optimal number is predicted by `information criteria' such as Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), and so on. On the other hand, the best performance is achieved by parameter tuning, when the data are produced from the same model as we hypothesized. We can tune the parameters of the model, according to the prescription such as the maximization of marginal likelihood. Both optimizations above appear to be different from each other. But they can be understood from a single point of view of `parameter scaling', which was shown in a pioneering work by Nemenman and Bialek [1] for density estimation problem. In this presentation, we show that the above two kinds of optimization are described in a unified form by a new type of information criteria related to the renormalization group equation. In the context of non-parametric models (e.g. [2]), we make clear the notions and roles of renormalization and renormalization group in Bayesian statistical inference, which lead to the improvement of learning performance of a broader class of models. References:\\ 1) I. Nemenman and W. Bialek, Phys.Rev.E. {\bf 65} (2002) 026137.\\ 2) W. Bialek, C.G. Callan and S.P. Strong, Phys.Rev.Lett. {\bf 77} (1996) 4693.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Aida, T.}, biburl = {https://www.bibsonomy.org/bibtex/2750fcc1468f25cca3cc4ce87e793e422/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {6ec11040984da2d5f0bf9fa56814106d}, intrahash = {750fcc1468f25cca3cc4ce87e793e422}, keywords = {bayesian criteria field group inference information models non-parametric renormalization statphys23 theory topic-11}, month = {9-13 July}, timestamp = {2007-06-20T10:16:19.000+0200}, title = {Renormalization and Renormalization Group in Non-parametric Bayesian Statistical Inference}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=404}, year = 2007 }