A vast number of questions and problems concerning probability theory need conditional probability for providing answers and solutions. From traditional games of dice to modern statistical...Show moreA vast number of questions and problems concerning probability theory need conditional probability for providing answers and solutions. From traditional games of dice to modern statistical applications and machine learning, all use conditional probability in some sense to obtain more insight in the problem. As fundamental conditional probability is, it is not without controversy. Problems and paradoxes like the Borel-Kolmogorov paradox, Monty Hall’s three door problem and the two envelope problem have puzzled mathematicians, statisticians and psychologists for centuries, resulting into much debate and a vast amount of literature. This thesis concerns some of the most well-known paradoxes in conditional probability. In all paradoxes, the paradoxical result arises from wrongly stating the probability space concerning the problem or wrongly applying conditional probability, like not giving the accompanying σ-algebra or not conditioning on a partition. All problems can be easily avoided by always stating the probability space with the σ-algebra when applying conditional probability. The two most obvious examples are the Borel-Kolmogorov paradox and Monty Hall’s problem. The Borel-Kolmogorov paradox is a good example of why conditioning on sets with zero measure is only possible with much care and why it is necessary to provide the accompanying σ-algebra with your solution. Monty Hall’s three door problem is a prime example of wrongly conditioning on a set of subsets that cannot form a partition of the sample space. The original problem asks for a single probability, however correctly applying conditional probability reveals that the probability of the car being behind the other door is dilated between the two values 1 2 to 1. In both cases the paradoxical results vanish when the whole probability space is considered and when conditional probability is applied correctly. The dilation of the conditional probability like in Monty Hall’s problem is investigated further in this thesis. Problems like Monty Hall and the boy or girl problem resemble each other in such a fundamental fashion that a generalization exists, encompassing them both. Furthermore, safe probability introduced by Grünwald [Grff18b] can be applied to answer the following question: if one should pin a single probability on for example in Monty Hall’s game the car being behind the other door, which probability should it be? This generalization can be applied to all problems with a countable space of outcomes with fixed probability measure and finite set of possible observations with sufficiently many possible probability measures, resolving several paradoxes in probability at once.Show less
Metastability is a phenomenon where a dynamical system can move between different states that are not its global equilibrium state. On short time scales the system can find itself equilibrized in a...Show moreMetastability is a phenomenon where a dynamical system can move between different states that are not its global equilibrium state. On short time scales the system can find itself equilibrized in a certain region of its state space (a local equilibrium), whereas on a long time scale it will make quick transitions between new, different regions of its state space. These local equilibria are referred to as the metastable states. One of the uses of metastability is for model reduction. In this thesis we will restrict ourselves to Markovian processes and consider the networks associated to the transitions of the Markov chains. Instead of considering a Markov process on a very large state space, one can look at the process on a reduced state space representing these metastable states. The idea is that this coarse-grained network "mimics" the behaviour of the original network. We shall give two different mathematical definitions for metastability of Markov chains. In most cases where metastability is studied, limiting asymptotics are wielded. One must think of taking limits of large volume or low temperature. However in the paper [1] by Avena, Castell, Gaudillière and Mélot a new framework is introduced by which to describe "metastability" without the use of these limits. The network of transitions of a given Markov process is coarse-grained to a state space that represents probability measures which focus on different regions of the original finite state space (the local equilibria). It does so through the use of intertwining dualities. We say that a n × n-matrix A is intertwined with a m × m-matrix C with respect to a m × n-matrix B if BA = CB. For our discussion we are given a Markov process on a finite state space with an associated transition matrix P in order to find another Markov process on a smaller state space with transition matrix P and a matrix Λ such that ΛP = PΛ where the rows of Λ are probability measures on the original state space (representing the local equilibria). In this thesis we will explore this framework based on intertwining on a toy model consisting of three nodes that we want to reduce to a network of two nodes. The goal is to illustrate the method in [1] in this explicit model and explore which evolutions among the local equilibria can be described; how this relates to the spectrum of the transition matrix P of the Markov chain in this model; and its implications on the mixing time.Show less
Gr¨obner bases are special sets of generators of ideals in multivariate polynomial rings, which provide some powerful theoretical and computational properties for solving problems in many fields...Show moreGr¨obner bases are special sets of generators of ideals in multivariate polynomial rings, which provide some powerful theoretical and computational properties for solving problems in many fields such as coding theory, cryptanalysis, optimization, geometric modelling, some problems in control theory, robotics, and statistics. One of the main ingredients in the construction of Gr¨obner bases is monomial orderings. In fact, monomial orderings have crucial role which determines the complexity of the computation of Gr¨obner bases. Different monomial orderings will lead to different levels of complexity in the computation process of a Gr¨obner basis. The first part of this thesis is devoted to discuss the well-known classification of monomial orderings by Robiano, which represents a monomial ordering on the set of monomials of a multivariate polynomial ring by a certain set of orthogonal vectors in R n . Furthermore, we give an explicit bijection between the set of all monomial orderings and such family of orthogonal vectors that enables us to construct a monomial ordering from such set of orthogonal vectors and vice versa. The first algorithm for computing Gr¨obner bases was formulated by Buchberger. The core of this algorithm is the concept of S-polynomials of pairs of polynomials. However, the major disadvantage of this algorithm is the amount of useless pairs of polynomials that the algorithm has to compute. Hence, the second part of this thesis is devoted to discuss the concept of Gr¨obner bases and Buchberger’s algorithm. Furthermore, we describe some strategies to optimize the computations of Gr¨obner bases. The last part of this thesis is devoted for the application of Gr¨obner bases in decoding problems. We describe how Gr¨obner bases get involved in solving decoding problems, especially for linear codes. Particularly, we discuss how to translate decoding problems of linear codes into the problem of solving system of multivariate polynomial equationsShow less
Multi-agent interactions are driven by both game dynamics and joint behavior. This implies that any specific agent faces a task in which the optimal policy is the best response to the opponents’...Show moreMulti-agent interactions are driven by both game dynamics and joint behavior. This implies that any specific agent faces a task in which the optimal policy is the best response to the opponents’ joint policy, which is not directly observable and must therefore be inferred from observations. Motivated by this setting, we propose a new formulation of deep state-space models (DSSMs) in multi-agent systems (MAS). This formulation results in models that can be used to represent the environment dynamics from an individual agent’s point of view, and ultimately to predict the other agents’ future actions. The models are built upon the framework of latent variable models, armed with the variational inference principle in regard to approximating the true (but intractable) posterior distributions. In addition, we explore extensions that make the model richer and thus more flexible by inducing covariance structures on the latent variable distributions. This leaves us with several models available, corresponding to the rank of the covariance inducing matrix. We then propose to choose the best-parsimonious model amongst them via the minimum description length (MDL) principle and illustrate the resulting performance on a given simple problem. Finally, we also demonstrate how the models explicitly affect the agent’s learned policy in a multi-agent reinforcement learning task. From the empirical implementation, we found that the models perform reasonably well to infer both the environment dynamics and the opponent’s policy. Overall, we believe that the proposed models are highly flexible, since the models can be used in any interaction setting of MAS (collaborative/competitive), partially observable environments, and any reinforcement learning paradigm (model-based/model-free), not to mention the models’ advantageous generative characteristics.Show less
In this work we describe the mathematical background of Generative Adversarial Networks (GANs). We discuss optimization criteria of the Jensen-Shannon GAN, the f-GAN, the Wasserstein GAN and the...Show moreIn this work we describe the mathematical background of Generative Adversarial Networks (GANs). We discuss optimization criteria of the Jensen-Shannon GAN, the f-GAN, the Wasserstein GAN and the Sobolev GAN. For the f-GAN we derive a variational formulation, which is closely related to the Wasserstein and Sobolev GAN criterion. The focus of the thesis will then shift towards Sobolev GANs. We show that optimizing the criterion belonging to these GANs can be reduced to a density estimation problem. The main result of this thesis is the convergence rate of Sobolev GANs when data lies on a simple manifold. The manifold used is a toy model and not realistic, but gives insight in the robustness of Sobolev GANs when data lies on a more general manifold. We use kernel density estimations to establish this convergence rate. When the function space over which the GAN is optimized is the L 2 -Sobolev space we found the following rates: for β ∈ R+ with 0 < β < d 2 we found the rate n −β d+3η , where η ∈ R+ is an arbitrary small number and d is the complete dimension of the observations and for β > d 2 we found the parametric rate n − 1 2 . The rates do not depend on the dimension d 0 of the manifold on which the data lies.Show less
In this thesis, we analyze and evaluate a phenomenological model for cell differentiation based on Hill-function type interaction kinetics. This is an extension of a model formulated by Dr. Sui...Show moreIn this thesis, we analyze and evaluate a phenomenological model for cell differentiation based on Hill-function type interaction kinetics. This is an extension of a model formulated by Dr. Sui Huang that has been proposed by Dr. Stefan Semrau to explain the observations of retinoic-acid driven mouse embryonic stem cells differentiation. Thereby, our main goal in this thesis is to evaluate the proposition that the model suffices as a conceptual mechanism of the performed experiments. Towards this end, we investigate how Frege’s theory of judgment can be used along with Kant’s theory of judgment to provide a systematic evaluation of phenomenological mathematical models.Show less
If 𝑚 denotes the number of digits in the regular continued fraction expansion that can be determined from 𝑛 digits in the decimal expansion, then Lochs’ Theorem states that the fraction 𝑚 𝑛...Show moreIf 𝑚 denotes the number of digits in the regular continued fraction expansion that can be determined from 𝑛 digits in the decimal expansion, then Lochs’ Theorem states that the fraction 𝑚 𝑛 converges Lebesgue almost surely to a fraction of two entropies as 𝑛 → ∞. These are the entropies of the interval maps that generate these expansions. Lochs’ Theorem has been generalized to pairs of interval maps that both belong to a class of piecewise monotonic transformations that generate expansions and that admit an invariant density with suitable ergodic properties. The first aim of this thesis is to review sufficient conditions on interval maps to belong to this class. For this, we first of all recover the famous existence result for invariant densities by Lasota and Yorke for expanding piecewise monotonic interval maps. As an example of a nonexpanding piecewise monotonic interval map, we also consider the Liverani-Saussol-Vaienti (LSV) map and provide a new proof of the already known result that such a map admits an invariant probability density if and only the corresponding parameter lies in (0, 1). Motivated by the practical use of beta encoders, one of the main goals in this thesis is to extend Lochs’ Theorem to expansions generated by a class of random piecewise monotonic interval maps. We review sufficient conditions on random interval maps to belong to this class. For two random interval maps 𝑇 and 𝑆 in this class, we show that, if 𝑚 denotes the number of digits in the 𝑆-expansion that can be determined from 𝑛 digits in the 𝑇-expansion, then, roughly speaking, the fraction 𝑚 𝑛 converges Lebesgue almost surely to a fraction of two fiber entropies as 𝑛 → ∞. As a second important goal, we prove that the skew product of an LSV map with parameter in (0, 1) and another LSV map with parameter in [1, ∞) and with underlying Bernoulli shift admits an invariant probability density.Show less
Many investors use optimization as a tool to make investment decisions. An investor decides which proportion of his wealth to invest in which asset class, thus composing his investment portfolio....Show moreMany investors use optimization as a tool to make investment decisions. An investor decides which proportion of his wealth to invest in which asset class, thus composing his investment portfolio. Given all asset classes available to the investor, with optimization the investor tries to find the portfolio with the most favourable risk and return trade-off. Historical data and models are used to estimate asset classes’ risks and returns. A problem with this approach is that optimal portfolios are often sensitive to variation in the uncertain risk- and return estimates, as shown in [18] and [23]. An alternative is to compute not only the optimal portfolio, but also the portfolios that are near-optimal, see [16]. This results in a near-optimal region of portfolios that can be shown to be more robust. The method to compute near-optimal regions is however computationally intensive. In this thesis, we reduce the computation time to determine a near-optimal region, without losing significant accuracy. When a near-optimal region is known, an investor still needs to decide which (near-optimal) portfolio to invest in. Near-optimal region estimates can be objects of high affine dimension, which are difficult to grasp and navigate for practitioners. In this thesis, we use polytope theory and show how this can be used to study the near-optimal region in more detail.Show less
