Lecture

Boltzmann Machines: Generating Synthetic Data in Finance

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Description

This lecture explores the application of Boltzmann machines in finance, focusing on generating synthetic data to address challenges like small datasets and data anonymization. The speaker explains the structure and operation of restricted Boltzmann machines, detailing how they learn and generate samples from the same distribution as the training data. The training process, involving stochastic gradient ascent and contrastive divergence, is discussed, along with the conversion of real-valued variables to binary representation. The lecture showcases the effectiveness of Boltzmann machines in learning and replicating the statistical properties of financial data, enabling the generation of synthetic data that closely matches the original dataset's characteristics.

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In information geometry, a divergence is a kind of statistical distance: a binary function which establishes the separation from one probability distribution to another on a statistical manifold. The simplest divergence is squared Euclidean distance (SED), and divergences can be viewed as generalizations of SED. The other most important divergence is relative entropy (also called Kullback–Leibler divergence), which is central to information theory.

Kullback–Leibler divergence

In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted , is a type of statistical distance: a measure of how one probability distribution P is different from a second, reference probability distribution Q. A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model when the actual distribution is P.

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In statistics, probability theory, and information theory, a statistical distance quantifies the distance between two statistical objects, which can be two random variables, or two probability distributions or samples, or the distance can be between an individual sample point and a population or a wider sample of points. A distance between populations can be interpreted as measuring the distance between two probability distributions and hence they are essentially measures of distances between probability measures.

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