Bernoulli distribution | EPFL Graph Search

In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with probability q = 1-p. Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have p \neq 1/2. The Bernoulli distribution is a special case of the binomial distribu

CauseOccam: Learning Interpretable Abstract Representations in Reinforcement Learning Environments via Model Sparsity

Sergei Volodin

"I choose this restaurant because they have vegan sandwiches" could be a typical explanation we would expect from a human. However, current Reinforcement Learning (RL) techniques are not able to provide such explanations, when trained on raw pixels. RL algorithms for state-of-the-art benchmark environments are based on neural networks, which lack interpretability, because of the very factor that makes them so versatile – they have many parameters and intermediate representations. Enforcing safety guarantees is important when deploying RL agents in the real world, and guarantees require interpretability of the agent. Humans use short explanations that capture only the essential parts and often contain few causes to explain an effect. In our thesis, we address the problem of making RL agents understandable by humans. In addition to the safety concerns, the quest to mimic human-like reasoning is of general scientific interest, as it sheds light on the easy problem of consciousness. The problem of providing interpretable and simple causal explanations of agent's behavior is connected to the problem of learning good state representations. If we lack such a representation, any reasoning algorithm's outputs would be useless for interpretability, since even the "referents" of the "thoughts" of such a system would be obscure to us. One way to define simplicity of causal explanations via the sparsity of the Causal Model that describes the environment: the causal graph has the fewest edges connecting causes to their effects. For example, a model for choosing the restaurant that only depends on the cause "vegan" is simpler and more interpretable than a model that looks at each pixel of a photo of the menu of a restaurant, and possibly relies as well on spurious correlations, such the style of the menu. In this thesis, we propose a framework "CauseOccam" for model-based Reinforcement Learning where the model is regularized for simplicity in terms of sparsity of the causal graph it corresponds to. The framework contains a learned mapping from observations to latent features, and a model predicting latent features at the next time-steps given ones from the current time-step. The latent features are regularized with the sparsity of the model, compared to a more traditional regularization on the features themselves, or via a hand-crafted interpretability loss. To achieve sparsity, we use discrete Bernoulli variables with gradient estimation, and to find the best parameters, we use the primal-dual constrained formulation to achieve a target model quality. The novelty of this work is in learning jointly a sparse causal graph and the representation taking pixels as the input on RL environments. We test this framework on benchmark environments with non-trivial high-dimensional dynamics and show that it can uncover the causal graph with the fewest edges in the latent space. We describe the implications of our work for developing priors enforcing interpretability.

2021

Timing is Everything

Karen Adam

In this thesis, timing is everything. In the first part, we mean this literally, as we tackle systems that encode information using timing alone. In the second part, we adopt the standard, metaphoric interpretation of this saying and show the importance of choosing the right time to sample a system, when efficiency is a key consideration.Time encoding machines, or, alternately, integrate-and-fire neurons, encode their inputs using spikes, the timings of which depend on the input and therefore hold information about it. These devices can be made more power efficient than their clocked counterparts and have thus been studied in the fields of signal processing, event-based vision, computational neuroscience and machine learning. However, their timing-based spiking output has so far often been considered a nuisance that one must make do with, rather than a potential advantage. In this thesis, we show that this timing-based output equips spiking devices with capabilities that are out of reach for classical encoding and processing systems.We first discover the benefits of time encoding on multi-channel encoding and recovery of a signal: with time encoding, clock alignment is easy to solve, although it poses problems in the classical sampling scenario. Then, we study the time encoding of low-dimensional signals and see that the asynchrony of spikes allows for a lower sample complexity in comparison with synchronous sampling. Thanks to this same asynchrony, time encoding of video results in an entanglement between spatial sampling density and temporal resolution---a relationship which is not present in frame-based video. Finally, we show that the all-or-none nature of the spikes allows training spiking neural networks in a layer-by-layer fashion---a feat that is impossible with clocked, artificial neural networks, due to the credit assignment problem.The second part of this thesis shows that choosing the right timing of samples can be crucial to ensure efficiency when performing nanoscale magnetic sensing. We are given a stochastic process, where each sample at time t follows a Bernoulli distribution, and which is characterized by oscillation frequencies that we are interested in recovering. We search for an optimal approach to sample this process, such that the variance of the frequencies' estimates is minimized, given constraints on the measurement time. The models we assume stem from the field of nanoscale magnetic sensing, where the number of parameters to be estimated varies with the number of spins one is trying to sense. We present an adaptive approach to choosing samples in both the single-spin and two-spin cases and compare the adaptive algorithm's performance to classical approaches to sampling.In both parts of the thesis, we move away from classical amplitude sampling and consider cases where timing takes the forefront and amplitude information is merely binary, to shows that timing can carry information and that it can control the amount of information gain.

EPFL2022

A new representation for multivariate tail probabilities

Jonathan A. Tawn, Jennifer Lynne Wadsworth

Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the effect of allowing the components to grow at different rates, and characterize the link between these marginal growth rates and the multivariate tail probability decay rate. Our approach leads to a whole class of univariate regular variation conditions, in place of the single but multivariate regular variation conditions that underpin the current theories. These conditions are indexed by a homogeneous function and an angular dependence function, which, for asymptotically independent random vectors, mirror the role played by the exponent measure and Pickands' dependence function in classical multivariate extremes. We additionally offer an inferential approach to joint survivor probability estimation. The key feature of our methodology is that extreme set probabilities can be estimated by extrapolating upon rays emanating from the origin when the margins of the variables are exponential. This offers an appreciable improvement over existing techniques where extrapolation in exponential margins is upon lines parallel to the diagonal.