Importance sampling the union of rare events with an application to power systems analysis

We consider importance sampling to estimate the probability $\mu$ of a union of $J$ rare events $H_j$ defined by a random variable $\boldsymbol{x}$. The sampler we study has been used in spatial statistics, genomics and combinatorics going back at least to Karp and Luby (1983). It works by sampling one event at random, then sampling $\boldsymbol{x}$ conditionally on that event happening and it constructs an unbiased estimate of $\mu$ by multiplying an inverse moment of the number of occuring events by the union bound. We prove some variance bounds for this sampler. For a sample size of $n$, it has a variance no larger than $\mu(\bar\mu-\mu)/n$ where $\bar\mu$ is the union bound. It also has a coefficient of variation no larger than $\sqrt{(J+J^{-1}-2)/(4n)}$ regardless of the overlap pattern among the $J$ events. Our motivating problem comes from power system reliability, where the phase differences between connected nodes have a joint Gaussian distribution and the $J$ rare events arise from unacceptably large phase differences. In the grid reliability problems even some events defined by $5772$ constraints in $326$ dimensions, with probability below $10^{-22}$, are estimated with a coefficient of variation of about $0.0024$ with only $n=10{,}000$ sample values

A probabilistic interpretation of set-membership filtering: application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zero-th order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P

Beyond Standard Assumptions - Semiparametric Models, A Dyadic Item Response Theory Model, and Cluster-Endogenous Random Intercept Models

In most statistical analyses, quantitative education researchers often make simplifying assumptions regarding the manner in which their data was generated in order to answer some of these questions. These assumptions can help to reduce the complexity of the problem, and allow the researcher to describe their data using a simpler, and often times more interpretable, statistical model. However, making some of these assumptions when they are not true can lead to biased estimates and misleading answers. While the standard sets of assumptions associated with commonly-used statistical models are usually sufficient in a wide range of contexts, it will always be beneficial for education researchers to understand what they are, when they are reasonable, and how to modify them if necessary. This dissertation focuses on three of the most common models used in quantitative education research (viz. parametric models like Linear Models (LMs), Item Response Theory (IRT) models, and Random-Intercept Models (RIMs)), discusses the standard sets of assumptions that accompany these models, and then describes related models with less stringent sets of assumptions. In each of the following three chapters, we either explicitly unpack existing models that are useful but are currently still uncommon in the field of education research, or propose novel models and/or estimation strategies for these models. We begin in Chapter 1 with a common parametric model known as the Gaussian LM, and use it as a scaffold to better understand semiparametric models and their estimation. We begin by reviewing how the coefficients of the Gaussian LM are usually estimated using Maximum Likelihood (ML) or Least-Squares (LS). We then introduce the notion of an $m$-estimator as well as that of a Regular Asymptotically Linear estimator, and show how they relate to the ML estimator. In particular, we introduce the notion of influence functions/curves and discuss their geometry together with concepts such as Hilbert spaces and tangent spaces. We then demonstrate, concretely, how to derive the so-called efficient influence function under the Gaussian LM, and show that it is precisely the influence function of the ML and (Ordinary) LS estimators. This shows that the ML estimator (at least under the Gaussian LM) is efficient. Using the foundation built, we move on from the Gaussian LM by relaxing both the assumption that the residuals are normally distributed, as well as the assumption that they have a constant variance, and define this as the Heteroskedastic Linear Model. Unlike the Gaussian LM, this is a semiparametric model. Where possible, we make use of intuition and analogous results from the parametric setting to help describe the workflow for obtaining an efficient estimator for the coefficients of the Heteroskedastic Linear Model. In particular, we derive the nuisance tangent space for this semiparametric model, and use it to obtain the efficient influence function for our model. We then show how to use the efficient influence function to obtain an efficient estimator (which happens to be the Weighted LS estimator) from the (Ordinary) LS estimator via a one-step approach as well as an estimating equations approach. We then conclude by directing readers to more advanced material, including references on more modern approaches to estimating more general semiparametric models such as Targeted Maximum Likelihood Estimation. In Chapter 2, we focus on a class of measurement models known as Item Response Theory models which are useful for measuring latent traits of a subject based on the subject's response to items. We relax the condition that the responses are only a result of the individual's latent trait (and possibly an external rater), and propose a dyadic Item Response Theory (dIRT) model for measuring interactions of pairs of individuals when the responses to items represent the actions (or behaviors, perceptions, etc.) of each individual (actor) made within the context of a dyad formed with another individual (partner). Examples of its use in education include the assessment of collaborative problem solving among students, or the evaluation of intra-departmental dynamics among teachers. The dIRT model generalizes both Item Response Theory models for measurement and the Social Relations Model for dyadic data. Here, the responses of an actor when paired with a partner are modeled as a function of not only the actor's inclination to act and the partner's tendency to elicit that action, but also the unique relationship of the pair, represented by two directional, possibly correlated, interaction latent variables. We discuss generalizations such as accommodating triads or larger groups, but focus on demonstrating the key idea in the dyadic case. We show that estimation may be performed using Markov-chain Monte Carlo implemented in \texttt{Stan}, making it straightforward to extend the dIRT model in various ways. Specifically, we show how the basic dIRT model can be extended to accommodate latent regressions, random effects, distal outcomes. We perform a simulation study that demonstrates that our estimation approach performs well. In the absence of educational data of this form, we demonstrate the usefulness of our proposed approach using speed-dating data instead, and find new evidence of pairwise interactions between participants, describing a mutual attraction that is inadequately characterized by individual properties alone.Finally, in Chapter 3, we consider the often implicit assumption made when estimating the coefficients of structural Random Intercept Models (RIMs) that covariates at all levels do not co-vary with the random intercepts. A violation of this assumption (called cluster-level endogeneity) leads to inconsistent estimates when using standard estimation procedures. For two-level RIMs with such endogeneity, Hausman and Taylor (HT) devised a consistent multi-step instrumental variable estimator using only internal instruments. We, instead, approach this problem by explicitly modeling the endogeneity using a Structural Equation Model (SEM). In this chapter, we compare, through simulation, the HT and SEM estimators, and evaluate their asymptotic and finite sample properties. We show that the SEM approach is also flexible enough to deal with different exchangeability assumptions for the covariates (e.g., whether the correlations between pairs of all units in a cluster are the same) and investigate how these exchangeability assumptions affect finite sample properties of the HT estimator. For the simulations, we propose a new procedure for generating cluster- and unit-level covariates and random intercepts with a fully flexible covariance structure. We also compare our approach to another common approach known as Multilevel Matching using data from the High School and Beyond survey

Modelling Learning as Modelling

Economists tend to represent learning as a procedure for estimating the parameters of the "correct" econometric model. We extend this approach by assuming that agents specify as well as estimate models. Learning thus takes the form of a dynamic process of developing models using an internal language of representation where expectations are formed by forecasting with the best current model. This introduces a distinction between the form and content of the internal models which is particularly relevant for boundedly rational agents. We propose a framework for such model development which use a combination of measures: the error with respect to past data, the complexity of the model, the cost of finding the model and a measure of the model's specificity The agent has to make various trade-offs between them. A utility learning agent is given as an example