Probabilité conditionnelle | EPFL Graph Search

vignette|Illustration des probabilités conditionnelles avec un diagramme d'Euler. On a la probabilité a priori P(A) = 0.30 + 0.10 + 0.12 = 0.52 et les probabilités conditionnelles P(A|B_1) = 1, P(A|B_2) = \frac{0.12}{0.12+0.04} = 0.75 et P(A|B_3) = 0.|320x320px En théorie des probabilités, une probabilité conditionnelle est la probabilité d'un événement sachant qu'un autre événement a eu lieu. Par exemple, si une carte d'un jeu est tirée au hasard, on estime qu'il y a une chance sur quatre d'obtenir un cœur ; mais si on aperçoit un reflet rouge sur la table, il y a maintenant une chance sur deux d'obtenir un cœur. Cette seconde estimation correspond à la probabilité conditionnelle d'obtenir un cœur sachant que la carte est rouge. Les probabilités conditionnelles font l'objet de paradoxes tels que le paradoxe des deux enfants, le paradoxe des deux enveloppes, le paradoxe des trois pièces de monnaie et le paradoxe des prisonniers.

Non-Exponential Variations for Classical Results in First Passage Percolation

Jacques Saliba

We study in this thesis the asymptotic behavior of optimal paths on a random graph model, the configuration model, for which we assign continuous random positive weights on its edges. We start by describing the asymptotic behavior of the diameter and the flooding time on the graph for a set of light-tailed edge-weights, and we prove that it is the largest class of densities for which these precise asymptotic expressions for the diameter/flooding time hold. We then show how the weighted optimal path can be constructed using a particular positive recurrent Markov chain. We finally study, in the last chapter, the noise sensitivity of the model by replacing every edge-weight independently by a new realization of the same distribution, with probability epsilon. We show that the optimal paths between two vertices before and after this modification are asymptotically ''independent'' conditioning on the graph, as the number of vertices tends to infinity.

EPFL2020

Financial Stability and the Macroeconomy

Corinne Dubois

The financial crisis of 2007-2009 drew attention to the essential role of banks for the macroeconomy and to the importance of having a resilient financial sector. A vulnerability in the financial sector spills over to the real economy and can drive it into a deep recession. Following the financial crisis, policy makers have issued rules and regulations aimed at strengthening the banks. My dissertation considers some of those measures, namely bank liquidity regulation and deposit insurance, and tries to assess their effectiveness in stabilizing the financial sector and the economy in general. The first chapter (joint with Prof. Luisa Lambertini) develops a model with regulated banks and a hedge fund to analyze the behavior of wholesale funding and the macroeconomic consequences of liquidity regulation. Banks raise deposits and subordinated wholesale funding from the hedge fund. Wholesale funding amplifies shocks: it is curtailed in economic downturns to avoid leveraging up and risk-taking by banks, further depressing credit and economic activity. By making banks safer, liquidity regulation increases wholesale funding and bank loans at the steady state. However, flat liquidity regulation requiring banks to hold a constant fraction of liquid assets can have unintended consequences and increase macroeconomic volatility. Our model further shows that cyclically adjusted regulation stabilizes the response of the economy to shocks. We also provide empirical evidence suggesting that liquidity helps contain the contraction in wholesale funding and loans during times of funding stress. The second chapter explores the macroeconomic and welfare implications of harmonizing or joining deposit insurance in the Eurozone. I develop a two-country model to represent the core and periphery countries in the Eurozone. The model features fragile banks that can be subject to endogenous and costly runs. The two countries are financially integrated through an international interbank market. I introduce deposit insurance and compare different insurance regimes in steady state and along the business cycle. The model emphasizes that financial integration implies co-movement in asset prices and the transmission of shocks across the border. I find that when deposit insurance has lower coverage in the periphery than in the core, financial stability is undermined in both regions due to higher asset price volatility and a higher probability of deposit runs. Hence, either harmonizing or joining deposit insurance leads to substantial welfare benefits for both countries. In the third chapter, I develop a macroeconomic model with multiple equilibria and bank runs to analyze deposit insurance. A bank run is a sunspot equilibrium, and it is caused by a self-fulfilling panic of all depositors. The model features a unique no-run equilibrium when macroeconomic conditions are good, but an unexpected negative shock may open up the possibility of a run equilibrium. A bank run is a rare but costly event that drives the economy in a long and severe recession. I analyze the introduction and build-up of a deposit insurance fund and show that it reduces the cost and probability of bank runs. The dynamic simulation of the model reveals that deposit insurance diminishes macroeconomic and financial instability following a technological shock.

EPFL2018

Reparameterizing Discontinuous Integrands for Differentiable Rendering

Wenzel Alban Jakob, Guillaume Rémi Bruno Loubet

Differentiable rendering has recently opened the door to a number of challenging inverse problems involving photorealistic images, such as computational material design and scattering-aware reconstruction of geometry and materials from photographs. Differentiable rendering algorithms strive to estimate partial derivatives of pixels in a rendered image with respect to scene parameters, which is difficult because visibility changes are inherently non-differentiable. We propose a new technique for differentiating path-traced images with respect to scene parameters that affect visibility, including the position of cameras, light sources, and vertices in triangle meshes. Our algorithm computes the gradients of illumination integrals by applying changes of variables that remove or strongly reduce the dependence of the position of discontinuities on differentiable scene parameters. The underlying parameterization is created on the fly for each integral and enables accurate gradient estimates using standard Monte Carlo sampling in conjunction with automatic differentiation. Importantly, our approach does not rely on sampling silhouette edges, which has been a bottleneck in previous work and tends to produce high-variance gradients when important edges are found with insufficient probability in scenes with complex visibility and high-resolution geometry. We show that our method only requires a few samples to produce gradients with low bias and variance for challenging cases such as glossy reflections and shadows. Finally, we use our differentiable path tracer to reconstruct the 3D geometry and materials of several real-world objects from a set of reference photographs.

ASSOC COMPUTING MACHINERY2019

Financial Stability and the Macroeconomy

Corinne Dubois

EPFL2018