Robert Gruhlke (FU Berlin)
Recent Advanced in Interacting particle methods for Bayesian Inference abstract
Abstract:
Ensemble methods have become ubiquitous for solving Bayesian inference problems, in particular the efficient sampling from posterior densities. State-of-the-art subclasses of Markow-Chain-Monte-Carlo methods rely on gradient information of the log-density including Langevin samplers such as Ensemble Kalman Sampler (EKS) and Affine Invariant Langevin Dynamics (ALDI). These dynamics are described by stochastic differential equations (SDEs) with time homogeneous drift terms. In this talk we present enhancement strategies of such ensemble methods based on sample enrichment and homotopy formalism, that ultimately lead to time-dependent drift terms that possible assimilate a larger class of target distributions while providing faster mixing times. Furthermore, we present an alternative route to construct time-inhomogeneous drift terms based on reverse Diffusion processes that are popular in state-of-the-art Generative Modelling such as Diffusion maps. Here, we propose learning these log-densities by propagation of the target distribution through an Ornstein-Uhlenbeck process. For this, we solve the associated Hamilton-Jabobi-Bellman equation through an adaptive explicit Euler discretization using low-rank compression such as functional Tensor Trains for the spatial discretization.
11:00 • Universität Basel, Spiegelgasse 5, Seminarraum 05.001
The negative Pell equation abstract
Abstract:
In this talk we will study the negative Pell equation, which is the conic C_D : x2 - D y2 = -1 to be solved in integers x, y in Z. We shall be concerned with the following question: as we vary over squarefree integers D, how often is C_D soluble? Stevenhagen conjectured an asymptotic formula for such D. Fouvry and Kluners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen\'s conjecture, and, time permitting, potential applications of the new proof techniques. This is joint work with Carlo Pagano.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Dr. Nadhir Ben Rached (University of Leeds, UK)
Importance Sampling for McKean-Vlasov Stochastic Differential Equation abstract
Abstract:
e are interested in Monte Carlo (MC) methods for estimating probabilities of rare events associated with solutions to the McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs arise in the mean-field limit of stochastic interacting particle systems, which have many applications in pedestrian dynamics, collective animal behaviour and financial mathematics. Importance sampling (IS) is used to reduce high relative variance in MC estimators of rare event probabilities. Optimal change of measure is methodically derived from variance minimisation, yielding a high-dimensional partial differential control equation which is cumbersome to solve. This problem is circumvented by using a decoupling approach, resulting in a lower dimensional control PDE. The decoupling approach necessitates the use of a double Loop Monte Carlo (DLMC) estimator. We further combine IS with a novel multilevel DLMC estimator which not only reduces complexity from O(TOL-4) to O(TOL-3) but also drastically reduces associated constant, enabling computationally feasible estimation of rare event probabilities. Joint work with Shyam Mohan, Abdul-Lateef Haji-Ali, and Raul Tempone.
14:15 • EPF Lausanne, CM 0 9
Zijian Guo (Rutgers University, USA)
Joint talk: Robust Causal Inference with Possibly Invalid Instruments: Post-selection Problems and A Solution Using Searching and Sampling abstract
Abstract:
Instrumental variable methods are among the most commonly used causal inference approaches to deal with unmeasured confounders in observational studies. The presence of invalid instruments is the primary concern for practical applications, and a fast-growing area of research is inference for the causal effect with possibly invalid instruments. This paper illustrates that the existing confidence intervals may undercover when the valid and invalid instruments are hard to separate in a data-dependent way. To address this, we construct uniformly valid confidence intervals that are robust to the mistakes in separating valid and invalid instruments. We propose to search for a range of treatment effect values that lead to sufficiently many valid instruments. We further devise a novel sampling method, which, together with searching, leads to a more precise confidence interval. Our proposed searching and sampling confidence intervals are uniformly valid and achieve the parametric length under the finite-sample majority and plurality rules. We apply our proposal to examine the effect of education on earnings. The proposed method is implemented in the R package \\texttt{RobustIV} available from CRAN.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Prof. Dr. Ernst Wit (University of Lugano)
Abstract:
Causality is the holy grail of science, but humankind has struggled to operationalize it for millennia. In recent decades, a number of more successful ways of dealing with causality in practice, such as propensity score matching, the PC algorithm, and invariant causal prediction, have been introduced. However, approaches that use a graphical model formulation tend to struggle with computational complexity, whenever the system gets large. Finding the causal structure typically becomes a combinatorial-hard problem.In our causal inference approach, we build forth on ideas present in invariant causal prediction and the causal Dantzig and anchor regression, by replacing combinatorial optimization with a continuous optimization using a form of causal regularization. This makes our method applicable to large systems. Furthermore, our approach allows a precise formulation of the trade-off between in-sample and out-of-sample prediction error.
15:15 • Universität Bern, IMSV, Alpeneggstrasse 22, 3012 Bern, Hörraum -203
Zijian Guo (Rutgers University, USA)
Joint talk: Robust Causal Inference with Possibly Invalid Instruments: Post-selection Problems and A Solution Using Searching and Sampling abstract
Abstract:
Instrumental variable methods are among the most commonly used causal inference approaches to deal with unmeasured confounders in observational studies. The presence of invalid instruments is the primary concern for practical applications, and a fast-growing area of research is inference for the causal effect with possibly invalid instruments. This paper illustrates that the existing confidence intervals may undercover when the valid and invalid instruments are hard to separate in a data-dependent way. To address this, we construct uniformly valid confidence intervals that are robust to the mistakes in separating valid and invalid instruments. We propose to search for a range of treatment effect values that lead to sufficiently many valid instruments. We further devise a novel sampling method, which, together with searching, leads to a more precise confidence interval. Our proposed searching and sampling confidence intervals are uniformly valid and achieve the parametric length under the finite-sample majority and plurality rules. We apply our proposal to examine the effect of education on earnings. The proposed method is implemented in the R package RobustIV available from CRAN.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Johanna Ziegel (Universität Bern)
Isotonic distributional regression abstract
Abstract:
Isotonic distributional regression (IDR) is a nonparametric distributional regression approach under a monotonicity constraint. It has found application as a generic method for uncertainty quantification, in statistical postprocessing of weather forecasts, and it is an integral part of distributional single index models. IDR has favorable calibration and optimality properties in finite samples.Furthermore, it has an interesting population counterpart called isotonic conditional laws that generalize conditional distributions with respect to σ-algebras to conditional distributions with respect to σ-lattices. In this talk, an overview of the theory and some applications of IDR are presented.
15:15 • EPF Lausanne MA A3 31
Dr. Michele Graffeo (Politecnico di Milano)
The geometry of double nested Hilbert shemes abstract
Abstract:
The Hilbert scheme Hilb^nX of n points on a quasi-projective variety X is a geometrical object introduced by Grothendieck and it has a prominent rôle in many areas of algebraic geometry. Recently, many variants of Hilb^nX have been introduced. My talk will focus on the double nested Hilbert scheme of points on X defined by S. Monavari. Specifically, I will explain how, when X is a smooth irreducible curve, its geometry is influenced by the combinatorics of reverse plane partitions and exhibits several pathologies.This is a joint project with Lella, Monavari, Ricolfi, Sammartano and it is partially funded by: PRIN 2020 “Squarefree Gröbner degenerations, special varieties and related topics” (MUR, project number 2020355B8Y)
16:00 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43