# Your search: "author:"Shahbaba, Babak""

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## Scholarly Works (48 results)

For an individual to successfully complete the task of decision-making, a set of temporally-organized events must occur: stimuli must be detected,

potential outcomes must be evaluated, behaviors must be executed or inhibited, and outcomes

(such as reward or punishment) must be experienced. Due to the complexity of this process,

it is very likely the case that decision-making is encoded by the temporally-precise interactions

among a population of neurons. Most existing statistical models, however, are inadequate for analyzing such sophisticated phenomenon as they either analyze a small number of neurons (e.g., pairwise analysis) or only provide an aggregated measure of interactions by assuming a constant dependence structure among neurons over time.

We start by proposing a scalable hierarchical semi-parametric Bayesian model to capture dependencies among multiple neurons by detecting their co-firing (possibly with some lag time). To this end, we model the spike train ( sequence of 1's (spike) and 0's (silence) ) for each neuron using the logistic function of a continuous latent variable with a Gaussian Process prior. Then we model the joint probability distribution of multiple neurons as a function of their corresponding marginal distribution using a parametric copula model. Our approach provides a flexible framework for modeling the underlying firing rates of each neuron. It also also allows us to make inference regarding both contemporaneous and lagged synchrony. We evaluate our approach using several simulation studies and apply it to analyze real data collected from an experiment designed for investigating the role of the prefrontal cortex of rats in reward-seeking behaviors.

Next, we propose a non-stationary Bayesian model to capture the dynamic nature of neuronal activity (such as the time-varying strength

of the interactions among neurons). Our proposed method yields results that provide new insights into the dynamic nature of population coding in the prefrontal cortex during decision making. In our analysis, we note that while some neurons in the prefrontal cortex do not synchronize their firing activity until the presence of a reward, a different set of neurons synchronize their

activity shortly after the onset of stimulus. These differentially synchronizing sub-populations of

neurons suggests a continuum of population representation of the reward-seeking task. Our analyses also suggest that the degree of synchronization differs between the

rewarded and non-rewarded conditions.

Finally we propose a novel statistical model for detecting neuronal communities involved in decision-making process. Our method characterizes the non-stationary activity of multiple neurons during a basic cognitive task by modeling their joint probability distribution dynamically. Our proposed model can capture the time-varying dependence structure among neurons while allowing the neuronal activity to change over time. This way, we are able to identify time-varying neuronal communities. By identifying communities of neurons that vary under different decisions, we expect our method to provide insights into the decision-making process in particular as well as into a broad range of cognitive functions.

This dissertation is an investigation into the intersections between differential geometry and Bayesian analysis. The former is the mathematical discipline that underlies our understanding of the spatial structure of the universe; the latter is the unified framework for statistical inference built upon the language of probability and the elegant Bayes' theorem. Here, the two disciplines are combined with the hope that a synergy might emerge and facilitate the useful application of Bayesian inference to real-world science. In particular, dynamic and high-dimensional neural data provides a challenging litmus test for the methods developed herein.

A major component of this work is the development and application of probabilistic models defined over smooth manifolds: dependencies between time series are modeled using the manifold of Hermitian positive definite matrices; probability density functions are modeled using the infinite sphere; and high-dimensional data are modeled using the Stiefel manifold of orthonormal matrices. Whereas formulating a manifold-based model is not difficult---in a certain sense, the geometry occurs a priori in each of the cases considered---the non-trivial geometry presents computational challenges for model-based inference. Hence, this thesis contributes two new algorithms for Bayesian inference on Riemannian manifolds. The first is an algorithm for inference over general Riemannian manifolds and is applied to inference on Hermitian positive definite matrices. The second is an algorithm for inference over manifolds that are embedded in Euclidean space and is applied to inference on the sphere and Stiefel manifolds.

This dissertation is ordered as follows. In Chapter 1, the general setting is introduced along with the rudiments of Riemannian geometry. In Chapter 2, the geodesic Lagrangian Monte Carlo algorithm is presented and used for Bayesian inference over the space of Hermitian positive definite matrices to learn the spectral densities of multivariate time series arising from local field potentials in a rodent brain. In Chapter 4, an alternative, conceptually simpler version of the geodesic Monte Carlo is developed, but the new algorithm requires differentiating the pseudo determinant, the derivative of which is derived in Chapter 3. In Chapter 5, the geometry of the infinite-dimensional sphere is leveraged for Bayesian nonparametric density estimation. In Chapter 6, high-dimensional spike trains and local field potentials in a rodent brain are used to predict environmental stimuli. This Bayesian `neural decoding' is facilitated by both geometric and non-geometric models. Chapter 7 charts the frontiers of Bayesian inference on infinite manifolds.

The availability of massive computational resources has led to a wide-spread application and development of Bayesian methods. However, in recent years, due to the explosive growth of data volume, developing advanced Bayesian methods for large-scale problems is still a very active area of research. This dissertation is an effort to develop more scalable computational tools for Bayesian inference in big data problems.

At its core, Bayesian inference involves evaluating high dimensional integrals with respect to the posterior distribution of model parameters and/or latent variables. However, the integration does not have closed form in general, and approximation methods are usually the only feasible option. Approximation can be divided into two main categories: deterministic approximation based on variational optimization, and stochastic approximation based on sampling methods.

We start with developing a new variational framework --- geometric approximation of posterior (GAP) --- based on ambient Fisher geometry. As a variational method, GAP has the potential to scale well to large problems compared to computationally expensive sampling methods. It not only has a well-established mathematical basis --- information geometry, but also works as a better alternative to other variational methods such as variational free energy and expectation propagation under certain scenarios.

Next, we focus on another class of approximation scheme based on MCMC sampling. Our method combines auto-encoders with Hamiltonian Monte Carlo (HMC). While HMC is efficient in exploring parameter space with high dimension or complicated geometry, it is computationally demanding since it has to evaluate additional geometric information of the parameter space. Our proposed method, Auto-encoding HMC, is designed to simulate Hamiltonian dynamics in a latent space with a much lower dimension, while still maintaining efficient exploration of the original space. Our method achieves a good balance between efficiency and accuracy for high-dimensional problems.

Besides our work on scalable approximation methods for Bayesian inference, we have also developed a variational auto-encoder (VAE) model based on determinantal point process (DPP) for big data classification problems with imbalanced classes. VAE is a generative model based on variational Bayes and is typically applied to high-dimensional data such as images and texts. In the presence of imbalanced data, our method balances the latent space by using a DPP prior to up-weight the minor classes. We successfully applied our method, henceforth called DPP-VAE, to neural data classification and hand-written digits generation, which are both high-dimensional in nature. Our method provides better results compared to standard VAE when datasets have imbalanced classes.

### Background

We investigate whether annotation of gene function can be improved using a classification scheme that is aware that functional classes are organized in a hierarchy. The classifiers look at phylogenic descriptors, sequence based attributes, and predicted secondary structure. We discuss three Bayesian models and compare their performance in terms of predictive accuracy. These models are the ordinary multinomial logit (MNL) model, a hierarchical model based on a set of nested MNL models, and an MNL model with a prior that introduces correlations between the parameters for classes that are nearby in the hierarchy. We also provide a new scheme for combining different sources of information. We use these models to predict the functional class of Open Reading Frames (ORFs) from the E. coli genome.### Results

The results from all three models show substantial improvement over previous methods, which were based on the C5 decision tree algorithm. The MNL model using a prior based on the hierarchy outperforms both the non-hierarchical MNL model and the nested MNL model. In contrast to previous attempts at combining the three sources of information in this dataset, our new approach to combining data sources produces a higher accuracy rate than applying our models to each data source alone.### Conclusion

Together, these results show that gene function can be predicted with higher accuracy than previously achieved, using Bayesian models that incorporate suitable prior information.Many model-based methods have been developed over the last several decades for analysis of multivariate time series, such as electroencephalograms (EEG) in order to understand electrical neural data. In this dissertation, we propose to use the functional boxplot to analyze log periodograms of EEG time series data in the spectral domain. The functional boxplot approach produces a median curve -- which is not equivalent to connecting medians obtained from frequency-specific boxplots. In addition, this approach identifies a functional median, summarizes variability and detects potential outliers. By extending functional boxplots analysis from one-dimensional curves to surfaces, surface boxplots are also used to explore the variation of the spectral power for the alpha ($8-12$ Hertz) and beta ($16-32$ Hertz) frequency bands across the brain cortical surface. By using rank-based nonparametric tests, we also investigate the stationarity of EEG traces across an exam acquired during resting-state by comparing the spectrum during the early vs. late phases of a single resting-state EEG exam.

Moreover, we present an exploratory data analysis tool for visualizing and testing the symmetric positive definite matrices (e.g., covariance, spectral and coherence matrices) in a multi--subject experimental setting. %derived from electroencephalogram (EEG) recordings from several clinical subjects. Our work is motivated by the clinician's interest to determine associations between brain functional connectivity (as measured by coherence) and patients' response to treatment. For each study participant, the geometric surface boxplot is developed to characterize the distribution of coherence matrices through the median matrix and the 50\% most central region of the data. The surface boxplot will also be used to detect the outlier coherence matrices as in the classical boxplot. To investigate the treatment effect, we develop a rank--based non--parametric approach to test for significant differences in coherence matrices between treatment and control groups. As an application, we demonstrate our proposed methods on coherence matrices, derived from an electroencephalograms, to determine potential associations treatment effect on patients with infantile spasms and hypsarrhythmia both before and after treatment.