Active Bayesian Optimization: Minimizing Minimizer Entropy

The ultimate goal of optimization is to find the minimizer of a target function.However, typical criteria for active optimization often ignore the uncertainty about the minimizer. We propose a novel criterion for global optimization and an associated sequential active learning strategy using Gaussian processes.Our criterion is the reduction of uncertainty in the posterior distribution of the function minimizer. It can also flexibly incorporate multiple global minimizers. We implement a tractable approximation of the criterion and demonstrate that it obtains the global minimizer accurately compared to conventional Bayesian optimization criteria

Bayesian Entropy Estimation for Countable Discrete Distributions

We consider the problem of estimating Shannon's entropy $H$ from discrete data, in cases where the number of possible symbols is unknown or even countably infinite. The Pitman-Yor process, a generalization of Dirichlet process, provides a tractable prior distribution over the space of countably infinite discrete distributions, and has found major applications in Bayesian non-parametric statistics and machine learning. Here we show that it also provides a natural family of priors for Bayesian entropy estimation, due to the fact that moments of the induced posterior distribution over $H$ can be computed analytically. We derive formulas for the posterior mean (Bayes' least squares estimate) and variance under Dirichlet and Pitman-Yor process priors. Moreover, we show that a fixed Dirichlet or Pitman-Yor process prior implies a narrow prior distribution over $H$, meaning the prior strongly determines the entropy estimate in the under-sampled regime. We derive a family of continuous mixing measures such that the resulting mixture of Pitman-Yor processes produces an approximately flat prior over $H$. We show that the resulting Pitman-Yor Mixture (PYM) entropy estimator is consistent for a large class of distributions. We explore the theoretical properties of the resulting estimator, and show that it performs well both in simulation and in application to real data.Comment: 38 pages LaTeX. Revised and resubmitted to JML

Bayesian entropy estimators for spike trains

Il Memming Park and Jonathan Pillow are with the Institute for Neuroscience and Department of Psychology, The University of Texas at Austin, TX 78712, USA -- Evan Archer is with the Institute for Computational and Engineering Sciences, The University of Texas at Austin, TX 78712, USA -- Jonathan Pillow is with the Division of Statistics and Scientific Computation, The University of Texas at Austin, Austin, TX 78712, USAPoster presentation: Information theoretic quantities have played a central role in neuroscience for quantifying neural codes [1]. Entropy and mutual information can be used to measure the maximum encoding capacity of a neuron, quantify the amount of noise, spatial and temporal functional dependence, learning process, and provide a fundamental limit for neural coding. Unfortunately, estimating entropy or mutual information is notoriously difficult--especially when the number of observations N is less than the number of possible symbols K [2]. For the neural spike trains, this is often the case due to the combinatorial nature of the symbols: for n simultaneously recorded neurons on m time bins, the number of possible symbols is K = 2n+m. Therefore, the question is how to extrapolate when you may have a severely under-sampled distribution. Here we describe a couple of recent advances in Bayesian entropy estimation for spike trains. Our approach follows that of Nemenman et al. [2], who formulated a Bayesian entropy estimator using a mixture-of-Dirichlet prior over the space of discrete distributions on K bins. We extend this approach to formulate two Bayesian estimators with different strategies to deal with severe under-sampling. For the first estimator, we design a novel mixture prior over countable distributions using the Pitman-Yor (PY) process [3]. The PY process is useful when the number of parameters is unknown a priori, and as a result finds many applications in Bayesian nonparametrics. PY process can model the heavy, power-law distributed tails which often occur in neural data. To reduce the bias of the estimator we analytically derive a set of mixing weights so that the resulting improper prior over entropy is approximately flat. We consider the posterior over entropy given a dataset (which contains some observed number of words but an unknown number of unobserved words), and show that the posterior mean can be efficiently computed via a simple numerical integral. The second estimator incorporates the prior knowledge about the spike trains. We use a simple Bernoulli process as a parametric model of the spike trains, and use a Dirichlet process to allow arbitrary deviation from the Bernoulli process. Under this model, very sparse spike trains are a priori orders of magnitude more likely than those with many spikes. Both estimators are computationally efficient, and statistically consistent. We applied those estimators to spike trains from early visual system to quantify neural coding [email protected]

Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed

Brain dynamics can exhibit narrow-band nonlinear oscillations and multistability. For a subset of disorders of consciousness and motor control, we hypothesize that some symptoms originate from the inability to spontaneously transition from one attractor to another. Using external perturbations, such as electrical pulses delivered by deep brain stimulation devices, it may be possible to induce such transition out of the pathological attractors. However, the induction of transition may be non-trivial, rendering the current open-loop stimulation strategies insufficient. In order to develop next-generation neural stimulators that can intelligently learn to induce attractor transitions, we require a platform to test the efficacy of such systems. To this end, we designed an analog circuit as a model for the multistable brain dynamics. The circuit spontaneously oscillates stably on two periods as an instantiation of a 3-dimensional continuous-time gated recurrent neural network. To discourage simple perturbation strategies such as constant or random stimulation patterns from easily inducing transition between the stable limit cycles, we designed a state-dependent nonlinear circuit interface for external perturbation. We demonstrate the existence of nontrivial solutions to the transition problem in our circuit implementation

Linear Time GPs for Inferring Latent Trajectories from Neural Spike Trains

Latent Gaussian process (GP) models are widely used in neuroscience to uncover hidden state evolutions from sequential observations, mainly in neural activity recordings. While latent GP models provide a principled and powerful solution in theory, the intractable posterior in non-conjugate settings necessitates approximate inference schemes, which may lack scalability. In this work, we propose cvHM, a general inference framework for latent GP models leveraging Hida-Mat\'ern kernels and conjugate computation variational inference (CVI). With cvHM, we are able to perform variational inference of latent neural trajectories with linear time complexity for arbitrary likelihoods. The reparameterization of stationary kernels using Hida-Mat\'ern GPs helps us connect the latent variable models that encode prior assumptions through dynamical systems to those that encode trajectory assumptions through GPs. In contrast to previous work, we use bidirectional information filtering, leading to a more concise implementation. Furthermore, we employ the Whittle approximate likelihood to achieve highly efficient hyperparameter learning.Comment: Published at ICML 202