Measuring graph similarity through continuous-time quantum walks and the quantum Jensen-Shannon divergence

In this paper we propose a quantum algorithm to measure the similarity between a pair of unattributed graphs. We design an experiment where the two graphs are merged by establishing a complete set of connections between their nodes and the resulting structure is probed through the evolution of continuous-time quantum walks. In order to analyze the behavior of the walks without causing wave function collapse, we base our analysis on the recently introduced quantum Jensen-Shannon divergence. In particular, we show that the divergence between the evolution of two suitably initialized quantum walks over this structure is maximum when the original pair of graphs is isomorphic. We also prove that under special conditions the divergence is minimum when the sets of eigenvalues of the Hamiltonians associated with the two original graphs have an empty intersection.

Attributed graph kernels using the Jensen-Tsallis q-differences

We propose a family of attributed graph kernels based on mutual information measures, i.e., the Jensen-Tsallis (JT) q-differences (for q  ∈ [1,2]) between probability distributions over the graphs. To this end, we first assign a probability to each vertex of the graph through a continuous-time quantum walk (CTQW). We then adopt the tree-index approach [1] to strengthen the original vertex labels, and we show how the CTQW can induce a probability distribution over these strengthened labels. We show that our JT kernel (for q  = 1) overcomes the shortcoming of discarding non-isomorphic substructures arising in the R-convolution kernels. Moreover, we prove that the proposed JT kernels generalize the Jensen-Shannon graph kernel [2] (for q = 1) and the classical subtree kernel [3] (for q = 2), respectively. Experimental evaluations demonstrate the effectiveness and efficiency of the JT kernels.

A continuous-time quantum walk kernel for unattributed graphs

Kernel methods provide a way to apply a wide range of learning techniques to complex and structured data by shifting the representational problem from one of finding an embedding of the data to that of defining a positive semidefinite kernel. In this paper, we propose a novel kernel on unattributed graphs where the structure is characterized through the evolution of a continuous-time quantum walk. More precisely, given a pair of graphs, we create a derived structure whose degree of symmetry is maximum when the original graphs are isomorphic. With this new graph to hand, we compute the density operators of the quantum systems representing the evolutions of two suitably defined quantum walks. Finally, we define the kernel between the two original graphs as the quantum Jensen-Shannon divergence between these two density operators. The experimental evaluation shows the effectiveness of the proposed approach. © 2013 Springer-Verlag.

Attributed graph similarity from the quantum Jensen-Shannon divergence

One of the most fundamental problem that we face in the graph domain is that of establishing the similarity, or alternatively the distance, between graphs. In this paper, we address the problem of measuring the similarity between attributed graphs. In particular, we propose a novel way to measure the similarity through the evolution of a continuous-time quantum walk. Given a pair of graphs, we create a derived structure whose degree of symmetry is maximum when the original graphs are isomorphic, and where a subset of the edges is labeled with the similarity between the respective nodes. With this compositional structure to hand, we compute the density operators of the quantum systems representing the evolution of two suitably defined quantum walks. We define the similarity between the two original graphs as the quantum Jensen-Shannon divergence between these two density operators, and then we show how to build a novel kernel on attributed graphs based on the proposed similarity measure. We perform an extensive experimental evaluation both on synthetic and real-world data, which shows the effectiveness the proposed approach. © 2013 Springer-Verlag.

Connecting social media to e-commerce:cold-start product recommendation using microblogging information

In recent years, the boundaries between e-commerce and social networking have become increasingly blurred. Many e-commerce websites support the mechanism of social login where users can sign on the websites using their social network identities such as their Facebook or Twitter accounts. Users can also post their newly purchased products on microblogs with links to the e-commerce product web pages. In this paper, we propose a novel solution for cross-site cold-start product recommendation, which aims to recommend products from e-commerce websites to users at social networking sites in 'cold-start' situations, a problem which has rarely been explored before. A major challenge is how to leverage knowledge extracted from social networking sites for cross-site cold-start product recommendation. We propose to use the linked users across social networking sites and e-commerce websites (users who have social networking accounts and have made purchases on e-commerce websites) as a bridge to map users' social networking features to another feature representation for product recommendation. In specific, we propose learning both users' and products' feature representations (called user embeddings and product embeddings, respectively) from data collected from e-commerce websites using recurrent neural networks and then apply a modified gradient boosting trees method to transform users' social networking features into user embeddings. We then develop a feature-based matrix factorization approach which can leverage the learnt user embeddings for cold-start product recommendation. Experimental results on a large dataset constructed from the largest Chinese microblogging service Sina Weibo and the largest Chinese B2C e-commerce website JingDong have shown the effectiveness of our proposed framework.

Mining product adopter information from online reviews for improving product recommendation

We present in this article an automated framework that extracts product adopter information from online reviews and incorporates the extracted information into feature-based matrix factorization formore effective product recommendation. In specific, we propose a bootstrapping approach for the extraction of product adopters from review text and categorize them into a number of different demographic categories. The aggregated demographic information of many product adopters can be used to characterize both products and users in the form of distributions over different demographic categories. We further propose a graphbased method to iteratively update user- and product-related distributions more reliably in a heterogeneous user-product graph and incorporate them as features into the matrix factorization approach for product recommendation. Our experimental results on a large dataset crawled from JINGDONG, the largest B2C e-commerce website in China, show that our proposed framework outperforms a number of competitive baselines for product recommendation.

Large scale data processing using MapReduce

As massive data sets become increasingly available, people are facing the problem of how to effectively process and understand these data. Traditional sequential computing models are giving way to parallel and distributed computing models, such as MapReduce, both due to the large size of the data sets and their high dimensionality. This dissertation, as in the same direction of other researches that are based on MapReduce, tries to develop effective techniques and applications using MapReduce that can help people solve large-scale problems. Three different problems are tackled in the dissertation. The first one deals with processing terabytes of raster data in a spatial data management system. Aerial imagery files are broken into tiles to enable data parallel computation. The second and third problems deal with dimension reduction techniques that can be used to handle data sets of high dimensionality. Three variants of the nonnegative matrix factorization technique are scaled up to factorize matrices of dimensions in the order of millions in MapReduce based on different matrix multiplication implementations. Two algorithms, which compute CANDECOMP/PARAFAC and Tucker tensor decompositions respectively, are parallelized in MapReduce based on carefully partitioning the data and arranging the computation to maximize data locality and parallelism.

Text analytics of social media: Sentiment analysis, event detection and summarization

In the last decade, large numbers of social media services have emerged and been widely used in people's daily life as important information sharing and acquisition tools. With a substantial amount of user-contributed text data on social media, it becomes a necessity to develop methods and tools for text analysis for this emerging data, in order to better utilize it to deliver meaningful information to users. ^ Previous work on text analytics in last several decades is mainly focused on traditional types of text like emails, news and academic literatures, and several critical issues to text data on social media have not been well explored: 1) how to detect sentiment from text on social media; 2) how to make use of social media's real-time nature; 3) how to address information overload for flexible information needs. ^ In this dissertation, we focus on these three problems. First, to detect sentiment of text on social media, we propose a non-negative matrix tri-factorization (tri-NMF) based dual active supervision method to minimize human labeling efforts for the new type of data. Second, to make use of social media's real-time nature, we propose approaches to detect events from text streams on social media. Third, to address information overload for flexible information needs, we propose two summarization framework, dominating set based summarization framework and learning-to-rank based summarization framework. The dominating set based summarization framework can be applied for different types of summarization problems, while the learning-to-rank based summarization framework helps utilize the existing training data to guild the new summarization tasks. In addition, we integrate these techneques in an application study of event summarization for sports games as an example of how to better utilize social media data. ^

Text Analytics of Social Media: Sentiment Analysis, Event Detection and Summarization

In the last decade, large numbers of social media services have emerged and been widely used in people's daily life as important information sharing and acquisition tools. With a substantial amount of user-contributed text data on social media, it becomes a necessity to develop methods and tools for text analysis for this emerging data, in order to better utilize it to deliver meaningful information to users. Previous work on text analytics in last several decades is mainly focused on traditional types of text like emails, news and academic literatures, and several critical issues to text data on social media have not been well explored: 1) how to detect sentiment from text on social media; 2) how to make use of social media's real-time nature; 3) how to address information overload for flexible information needs. In this dissertation, we focus on these three problems. First, to detect sentiment of text on social media, we propose a non-negative matrix tri-factorization (tri-NMF) based dual active supervision method to minimize human labeling efforts for the new type of data. Second, to make use of social media's real-time nature, we propose approaches to detect events from text streams on social media. Third, to address information overload for flexible information needs, we propose two summarization framework, dominating set based summarization framework and learning-to-rank based summarization framework. The dominating set based summarization framework can be applied for different types of summarization problems, while the learning-to-rank based summarization framework helps utilize the existing training data to guild the new summarization tasks. In addition, we integrate these techneques in an application study of event summarization for sports games as an example of how to better utilize social media data.

Sobre a integração indefinida de funções racionais complexas: teoria e implementação de algoritmos racionais

We present indefinite integration algorithms for rational functions over subfields of the complex numbers, through an algebraic approach. We study the local algorithm of Bernoulli and rational algorithms for the class of functions in concern, namely, the algorithms of Hermite; Horowitz-Ostrogradsky; Rothstein-Trager and Lazard-Rioboo-Trager. We also study the algorithm of Rioboo for conversion of logarithms involving complex extensions into real arctangent functions, when these logarithms arise from the integration of rational functions with real coefficients. We conclude presenting pseudocodes and codes for implementation in the software Maxima concerning the algorithms studied in this work, as well as to algorithms for polynomial gcd computation; partial fraction decomposition; squarefree factorization; subresultant computation, among other side algorithms for the work. We also present the algorithm of Zeilberger-Almkvist for integration of hyperexpontential functions, as well as its pseudocode and code for Maxima. As an alternative for the algorithms of Rothstein-Trager and Lazard-Rioboo-Trager, we yet present a code for Benoulli’s algorithm for square-free denominators; and another for Czichowski’s algorithm, although this one is not studied in detail in the present work, due to the theoretical basis necessary to understand it, which is beyond this work’s scope. Several examples are provided in order to illustrate the working of the integration algorithms in this text

Optimización de la factorización de matrices no negativas en Bioinformática

En los últimos años se ha incrementado el interés de la comunidad científica en la Factorización de matrices no negativas (Non-negative Matrix Factorization, NMF). Este método permite transformar un conjunto de datos de grandes dimensiones en una pequeña colección de elementos que poseen semántica propia en el contexto del análisis. En el caso de Bioinformática, NMF suele emplearse como base de algunos métodos de agrupamiento de datos, que emplean un modelo estadístico para determinar el número de clases más favorable. Este modelo requiere de una gran cantidad de ejecuciones de NMF con distintos parámetros de entrada, lo que representa una enorme carga de trabajo a nivel computacional. La mayoría de las implementaciones de NMF han ido quedando obsoletas ante el constante crecimiento de los datos que la comunidad científica busca analizar, bien sea porque los tiempos de cómputo llegan a alargarse hasta convertirse en inviables, o porque el tamaño de esos datos desborda los recursos del sistema. Por ello, esta tesis doctoral se centra en la optimización y paralelización de la factorización NMF, pero no solo a nivel teórico, sino con el objetivo de proporcionarle a la comunidad científica una nueva herramienta para el análisis de datos de origen biológico. NMF expone un alto grado de paralelismo a nivel de datos, de granularidad variable; mientras que los métodos de agrupamiento mencionados anteriormente presentan un paralelismo a nivel de cómputo, ya que las diversas instancias de NMF que se ejecutan son independientes. Por tanto, desde un punto de vista global, se plantea un modelo de optimización por capas donde se emplean diferentes tecnologías de alto rendimiento...

Enhancing non-refractory aerosol apportionment from an urban industrial site through receptor modelling of complete high time-resolution aerosol mass spectra

Receptor modelling was performed on quadrupole unit mass resolution aerosol mass spectrometer (Q-AMS) sub-micron particulate matter (PM) chemical speciation measurements from Windsor, Ontario, an industrial city situated across the Detroit River from Detroit, Michigan. Aerosol and trace gas measurements were collected on board Environment Canada’s CRUISER mobile laboratory. Positive matrix factorization (PMF) was performed on the AMS full particle-phase mass spectrum (PMFFull MS) encompassing both organic and inorganic components. This approach was compared to the more common method of analysing only the organic mass spectra (PMFOrg MS). PMF of the full mass spectrum revealed that variability in the non-refractory sub-micron aerosol concentration and composition was best explained by six factors: an amine-containing factor (Amine); an ammonium sulphate and oxygenated organic aerosol containing factor (Sulphate-OA); an ammonium nitrate and oxygenated organic aerosol containing factor (Nitrate-OA); an ammonium chloride containing factor (Chloride); a hydrocarbon like organic aerosol (HOA) factor; and a moderately oxygenated organic aerosol factor (OOA). PMF of the organic mass spectrum revealed three factors of similar composition to some of those revealed through PMFFull MS: Amine, HOA and OOA. Including both the inorganic and organic mass proved to be a beneficial approach to analysing the unit mass resolution AMS data for several reasons. First, it provided a method for potentially calculating more accurate sub-micron PM mass concentrations, particularly when unusual factors are present, in this case, an Amine factor. As this method does not rely on a priori knowledge of chemical species, it circumvents the need for any adjustments to the traditional AMS species fragmentation patterns to account for atypical species, and can thus lead to more complete factor profiles. It is expected that this method would be even more useful for HR-ToF-AMS data, due to the ability to better understand the chemical nature of atypical factors from high resolution mass spectra. Second, utilizing PMF to extract factors containing inorganic species allowed for the determination of extent of neutralization, which could have implications for aerosol parameterization. Third, subtler differences in organic aerosol components were resolved through the incorporation of inorganic mass into the PMF matrix. The additional temporal features provided by the inorganic aerosol components allowed for the resolution of more types of oxygenated organic aerosol than could be reliably re-solved from PMF of organics alone. Comparison of findings from the PMFFull MS and PMFOrg MS methods showed that for the Windsor airshed, the PMFFull MS method enabled additional conclusions to be drawn in terms of aerosol sources and chemical processes. While performing PMFOrg MS can provide important distinctions between types of organic aerosol, it is shown that including inorganic species in the PMF analysis can permit further apportionment of organics for unit mass resolution AMS mass spectra.

Characterization of urban aerosol in Cork city (Ireland) using aerosol mass spectrometry

Ambient wintertime background urban aerosol in Cork city, Ireland, was characterized using aerosol mass spectrometry. During the three-week measurement study in 2009, 93% of the ca. 1 350 000 single particles characterized by an Aerosol Time-of-Flight Mass Spectrometer (TSI ATOFMS) were classified into five organic-rich particle types, internally mixed to different proportions with elemental carbon (EC), sulphate and nitrate, while the remaining 7% was predominantly inorganic in nature. Non-refractory PM1 aerosol was characterized using a High Resolution Time-of-Flight Aerosol Mass Spectrometer (Aerodyne HR-ToF-AMS) and was also found to comprise organic aerosol as the most abundant species (62 %), followed by nitrate (15 %), sulphate (9 %) and ammonium (9 %), and chloride (5 %). Positive matrix factorization (PMF) was applied to the HR-ToF-AMS organic matrix, and a five-factor solution was found to describe the variance in the data well. Specifically, "hydrocarbon-like" organic aerosol (HOA) comprised 20% of the mass, "low-volatility" oxygenated organic aerosol (LV-OOA) comprised 18 %, "biomass burning" organic aerosol (BBOA) comprised 23 %, non-wood solid-fuel combustion "peat and coal" organic aerosol (PCOA) comprised 21 %, and finally a species type characterized by primary m/z peaks at 41 and 55, similar to previously reported "cooking" organic aerosol (COA), but possessing different diurnal variations to what would be expected for cooking activities, contributed 18 %. Correlations between the different particle types obtained by the two aerosol mass spectrometers are also discussed. Despite wood, coal and peat being minor fuel types used for domestic space heating in urban areas, their relatively low combustion efficiencies result in a significant contribution to PM1 aerosol mass (44% and 28% of the total organic aerosol mass and non-refractory total PM1, respectively).Ambient wintertime background urban aerosol in Cork city, Ireland, was characterized using aerosol mass spectrometry. During the three-week measurement study in 2009, 93% of the ca. 1 350 000 single particles characterized by an Aerosol Time-of-Flight Mass Spectrometer (TSI ATOFMS) were classified into five organic-rich particle types, internally mixed to different proportions with elemental carbon (EC), sulphate and nitrate, while the remaining 7% was predominantly inorganic in nature. Non-refractory PM1 aerosol was characterized using a High Resolution Time-of-Flight Aerosol Mass Spectrometer (Aerodyne HR-ToF-AMS) and was also found to comprise organic aerosol as the most abundant species (62 %), followed by nitrate (15 %), sulphate (9 %) and ammonium (9 %), and chloride (5 %). Positive matrix factorization (PMF) was applied to the HR-ToF-AMS organic matrix, and a five-factor solution was found to describe the variance in the data well. Specifically, "hydrocarbon-like" organic aerosol (HOA) comprised 20% of the mass, "low-volatility" oxygenated organic aerosol (LV-OOA) comprised 18 %, "biomass burning" organic aerosol (BBOA) comprised 23 %, non-wood solid-fuel combustion "peat and coal" organic aerosol (PCOA) comprised 21 %, and finally a species type characterized by primary m/z peaks at 41 and 55, similar to previously reported "cooking" organic aerosol (COA), but possessing different diurnal variations to what would be expected for cooking activities, contributed 18 %. Correlations between the different particle types obtained by the two aerosol mass spectrometers are also discussed. Despite wood, coal and peat being minor fuel types used for domestic space heating in urban areas, their relatively low combustion efficiencies result in a significant contribution to PM1 aerosol mass (44% and 28% of the total organic aerosol mass and non-refractory total PM1, respectively).

Single particle characterization using the soot particle aerosol mass spectrometer (SP-AMS)

Understanding the impact of atmospheric black carbon (BC) containing particles on human health and radiative forcing requires knowledge of the mixing state of BC, including the characteristics of the materials with which it is internally mixed. In this study, we demonstrate for the first time the capabilities of the Aerodyne Soot-Particle Aerosol Mass Spectrometer equipped with a light scattering module (LS-SP-AMS) to examine the mixing state of refractory BC (rBC) and other aerosol components in an urban environment (downtown Toronto). K-means clustering analysis was used to classify single particle mass spectra into chemically distinct groups. One resultant cluster is dominated by rBC mass spectral signals (C+1 to C+5) while the organic signals fall into a few major clusters, identified as hydrocarbon-like organic aerosol (HOA), oxygenated organic aerosol (OOA), and cooking emission organic aerosol (COA). A nearly external mixing is observed with small BC particles only thinly coated by HOA ( 28% by mass on average), while over 90% of the HOA-rich particles did not contain detectable amounts of rBC. Most of the particles classified into other inorganic and organic clusters were not significantly associated with BC. The single particle results also suggest that HOA and COA emitted from anthropogenic sources were likely major contributors to organic-rich particles with low to mid-range aerodynamic diameter (dva). The similar temporal profiles and mass spectral features of the organic clusters and the factors from a positive matrix factorization (PMF) analysis of the ensemble aerosol dataset validate the conventional interpretation of the PMF results.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.