Exact Non-Parametric Bayesian Inference on Infinite Trees

Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, moments, and other quantities. We prove asymptotic convergence and consistency results, and illustrate the behavior of our model on some prototypical functions.Comment: 32 LaTeX pages, 9 figures, 5 theorems, 1 algorith

Fast non-parametric Bayesian inference on infinite trees

Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, and other quantities

Fast Non-Parametric Bayesian Inference on Infinite Trees

Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, and other quantities.Comment: 8 twocolumn pages, 3 figure

Dynkin operators and renormalization group actions in pQFT

Renormalization techniques in perturbative quantum field theory were known, from their inception, to have a strong combinatorial content emphasized, among others, by Zimmermann's celebrated forest formula. The present article reports on recent advances on the subject, featuring the role played by the Dynkin operators (actually their extension to the Hopf algebraic setting) at two crucial levels of renormalization, namely the Bogolioubov recursion and the renormalization group (RG) equations. For that purpose, an iterated integrals toy model is introduced to emphasize how the operators appear naturally in the setting of renormalization group analysis. The toy model, in spite of its simplicity, captures many key features of recent approaches to RG equations in pQFT, including the construction of a universal Galois group for quantum field theories

Analytical two-center integrals over Slater geminal functions

We present analytical formulas for the calculation of the two-center two-electron integrals in the basis of Slater geminals and products of Slater orbitals. Our derivation starts with establishing a inhomogeneous fourth-order ordinary differential equation that is obeyed by the master integral, the simplest integral with inverse powers of all interparticle distances. To solve this equation it was necessary to introduce a new family of special functions which are defined through their series expansions around regular singular points of the differential equation. To increase the power of the interparticle distances under the sign of the integral we developed a family of open-ended recursion relations. A handful of special cases of the integrals is also analysed with some remarks on simplifications that occur. Additionally, we present some numerical examples of the master integral that validate the usefulness and correctness of the key equations derived in this paper. In particular, we compare our results with the calculations based on the series expansion of the exp(-\gamma r12) term in the master integral.Comment: 28 pages, 0 figures, 7 table

Fredholm Determinants, Differential Equations and Matrix Models

Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. There is also an exponential variant of this analysis which includes the circular ensembles of NxN unitary matrices.Comment: 34 pages, LaTeX using RevTeX 3.0 macros; last version changes only the abstract and decreases length of typeset versio

Inversion of the star transform

We define the star transform as a generalization of the broken ray transform introduced by us in previous work. The advantages of using the star transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium separately and simultaneously (from the same data) and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is discarded. In this paper, we derive the star transform from physical principles, discuss its mathematical properties and analyze numerical stability of inversion. In particular, it is shown that stable inversion of the star transform can be obtained only for configurations involving odd number of rays. Several computationally-efficient inversion algorithms are derived and tested numerically.Comment: Accepted to Inverse Problems in this for

WKB solutions of difference equations and reconstruction by the topological recursion

The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a $\hbar$-difference equation: $\Psi(x+\hbar)=\left(e^{\hbar\frac{d}{dx}}\right) \Psi(x)=L(x;\hbar)\Psi(x)$ with $L(x;\hbar)\in GL_2( (\mathbb{C}(x))[\hbar])$. In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of $\hbar$-differential systems to this setting. We apply our results to a specific $\hbar$-difference system associated to the quantum curve of the Gromov-Witten invariants of $\mathbb{P}^1$ for which we are able to prove that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve $y=\cosh^{-1}\frac{x}{2}$. Finally, identifying the large $x$ expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of $\mathbb{P}^1$.Comment: 41 pages, 2 figures, published version in Nonlinearit