A population model-based linear-quadratic Gaussian compensator for the control of intravenously infused alcohol studies and withdrawal symptom prophylaxis using transdermal sensing.

An output feedback LQG compensator (combined controller and state estimator) for the regulation of intravenous-infused alcohol studies and treatment using a noninvasive transdermal alcohol biosensor is developed. The design is based on a population model involving an abstract semi-linear parabolic hybrid reaction-diffusion system involving coupled partial and ordinary differential equations with random parameters known only up to their distributions. The scheme developed is based on a weak formulation of the model equations in an appropriately constructed Gelfand triple of Bochner spaces wherein the unknown random parameters are treated as additional spatial variables. Implementation relies on a Galerkin-based approximation and convergence theory and an abstract formulation involving linear semigroups of operators. The model is fit and validated using laboratory collected human subject data and the method of moments. The results of numerical simulations of controlled intravenous alcohol infusion are presented and discussed.

Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol.

The existence and consistency of a maximum likelihood estimator for the joint probability distribution of random parameters in discrete-time abstract parabolic systems was established by taking a nonparametric approach in the context of a mixed effects statistical model using a Prohorov metric framework on a set of feasible measures. A theoretical convergence result for a finite dimensional approximation scheme for computing the maximum likelihood estimator was also established and the efficacy of the approach was demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic partial differential equation (PDE). Numerical studies included show that the maximum likelihood estimator is statistically consistent, demonstrated by the convergence of the estimated distribution to the "true" distribution in an example involving simulated data. The algorithm developed was then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation (LOOCV) method, we found an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode was then used to quantify the uncertainty propagated from the random parameters to the output of the model in the form of a 95 error band surrounding the estimated output signal.

Tracking and blind deconvolution of blood alcohol concentration from transdermal alcohol biosensor data: A population model-based LQG approach in Hilbert space.

LQG control in Hilbert space, a novel approach for random abstract parabolic systems, and new transdermal alcohol biosensor technology are combined to yield tracking controllers that can be used to automate inpatient management of alcohol withdrawal syndrome and human subject intravenous alcohol infusion studies, and to blindly deconvolve blood or breath alcohol concentration from biosensor measured transdermal alcohol level. The approach taken is based on a full-body alcohol population model in the form of a random, nonlinear, hybrid system of ordinary and partial differential equations and its abstract formulation in a Gelfand triple of Bochner spaces. The efficacy of the approach is demonstrated through simulation studies based on laboratory collected drinking data.

Social anxiety and alcohol consumption: The role of social context.

Research has identified social anxiety as a risk factor for the development of alcohol use disorder. However, studies have produced equivocal findings regarding the relationship between social anxiety and drinking behaviors in authentic drinking environments. This study examined how social-contextual features of real-world drinking contexts might influence the relationship between social anxiety and alcohol consumption in everyday settings. At an initial laboratory visit, heavy social drinkers (N = 48) completed the Liebowitz Social Anxiety Scale. Participants were then outfitted with a transdermal alcohol monitor individually-calibrated for each participant via laboratory alcohol-administration. Over the next seven days, participants wore this transdermal alcohol monitor and responded to random survey prompts (6x/day), during which they provided photographs of their surroundings. Participants then reported on their levels of social familiarity with individuals visible in photographs. Multilevel models indicated a significant interaction between social anxiety and social familiarity in predicting drinking, b = -0.004, p =.003 Specifically, among participants higher in social anxiety, drinking increased as social familiarity decreased b = -0.152, p <.001, whereas among those lower in social anxiety, this relationship was non-significant, b = 0.007, p =.867. Considered alongside prior research, findings suggest that the presence of strangers within a given environment may play a role in the drinking behavior of socially anxious individuals.

Uncertainty Quantification in Estimating Blood Alcohol Concentration From Transdermal Alcohol Level With Physics-Informed Neural Networks.

We develop an approach to estimate a blood alcohol signal from a transdermal alcohol signal using physics-informed neural networks (PINNs). Specifically, we use a generative adversarial network (GAN) with a residual-augmented loss function to estimate the distribution of unknown parameters in a diffusion equation model for transdermal transport of alcohol in the human body. We design another PINN for the deconvolution of the blood alcohol signal from the transdermal alcohol signal. Based on the distribution of the unknown parameters, this network is able to estimate the blood alcohol signal and quantify the uncertainty in the form of conservative error bands. Finally, we show how a posterior latent variable can be used to sharpen these conservative error bands. We apply the techniques to an extensive dataset of drinking episodes and demonstrate the advantages and shortcomings of this approach.

An Abstract Parabolic System-Based Physics-Informed Long Short-Term Memory Network for Estimating Breath Alcohol Concentration from Transdermal Alcohol Biosensor Data.

The problem of estimating breath alcohol concentration based on transdermal alcohol biosensor data is considered. Transdermal alcohol concentration provides a promising alternative to classical methods such as breathalyzers or drinking diaries. A physics-informed long Short-term memory (LSTM) network with covariates for the solution of the estimation problem is developed. The data-driven nature of an LSTM is augmented with a first principles physics-based population model for the diffusion of ethanol through the epidermal layer of the skin. The population model in an abstract parabolic framework appears as part of a regularization term in the loss function of the LSTM. While learning, the model is encouraged to both fit the data and to produce physically meaningful outputs. To deal with the high variation observed in the data, a mechanism for the uncertainty quantification of the estimates based on a recently discovered relation between Monte-Carlo dropout and Bayesian learning is used. The physics-based population model and the LSTM are trained and tested using controlled laboratory collected breath and transdermal alcohol data collected in four sessions from 40 orally dosed participants (50% female, ages 21 - 33 years, 35% BMI above 25.0) resulting in 256 usable drinking episodes partitioned into training and testing sets. Body measurement (e.g. BMI, hip to waist ratio, etc.), personal (e.g. sex, age, race, etc.), drinking behavior (e.g. frequent, rarely, etc.), and environmental (e.g. temperature, humidity, etc.) covariates were also collected from participants. The importance of various covariates in the estimation is investigated using Shapley values. It is shown that the physics-informed LSTM network can be successfully applied to drinking episodes from both the training and test set, and that the physics-based information leads to better generalization ability on new drinking episodes with the uncertainty quantification yielding credible bands that effectively capture the true signal. Compared to two machine learning models from previous studies, the proposed model reduces relative L2 error in estimated breath alcohol concentration by 58% and 72%, and relative peak error by 33% and 76%.

Blood and Breath Alcohol Concentration from Transdermal Alcohol Biosensor Data: Estimation and Uncertainty Quantification via Forward and Inverse Filtering for a Covariate-Dependent, Physics-Informed, Hidden Markov Model.

Transdermal alcohol biosensors that do not require active participation of the subject and yield near continuous measurements have the potential to significantly enhance the data collection abilities of alcohol researchers and clinicians who currently rely exclusively on breathalyzers and drinking diaries. Making these devices accessible and practical requires that transdermal alcohol concentration (TAC) be accurately and consistently transformable into the well-accepted measures of intoxication, blood/breath alcohol concentration (BAC/BrAC). A novel approach to estimating BrAC from TAC based on covariate-dependent physics-informed hidden Markov models with two emissions is developed. The hidden Markov chain serves as a forward full-body alcohol model with BrAC and TAC, the two emissions, assumed to be described by a bivariate normal which depends on the hidden Markovian states and person-level and session-level covariates via built-in regression models. An innovative extension of hidden Markov modeling is developed wherein the hidden Markov model framework is regularized by a first-principles PDE model to yield a hybrid that combines prior knowledge of the physics of transdermal ethanol transport with data-based learning. Training, or inverse filtering, is effected via the Baum-Welch algorithm and 256 sets of BrAC and TAC signals and covariate measurements collected in the laboratory. Forward filtering of TAC to obtain estimated BrAC is achieved via a new physics-informed regularized Viterbi algorithm which determines the most likely path through the hidden Markov chain using TAC alone. The Markovian states are decoded and used to yield estimates of BrAC and to quantify the uncertainty in the estimates. Numerical studies are presented and discussed. Overall good agreement between BrAC data and estimates was observed with a median relative peak error of 22% and a median relative area under the curve error of 25% on the test set. We also demonstrate that the physics-informed Viterbi algorithm eliminates non-physical artifacts in the BrAC estimates.

Deconvolving breath alcohol concentration from biosensor measured transdermal alcohol level under uncertainty: a Bayesian approach.

The posterior distribution (PD) of random parameters in a distributed parameter-based population model for biosensor measured transdermal alcohol is estimated. The output of the model is transdermal alcohol concentration (TAC), which, via linear semigroup theory can be expressed as the convolution of blood or breath alcohol concentration (BAC or BrAC) with a filter that depends on the individual participant or subject, the biosensor hardware itself, and environmental conditions, all of which can be considered to be random under the presented framework. The distribution of the input to the model, the BAC or BrAC, is also sequentially estimated. A Bayesian approach is used to estimate the PD of the parameters conditioned on the population sample's measured BrAC and TAC. We then use the PD for the parameters together with a weak form of the forward random diffusion model to deconvolve an individual subject's BrAC conditioned on their measured TAC. Priors for the model are obtained from simultaneous temporal population observations of BrAC and TAC via deterministic or statistical methods. The requisite computations require finite dimensional approximation of the underlying state equation, which is achieved through standard finite element (i.e., Galerkin) techniques. The posteriors yield credible regions, which remove the need to calibrate the model to every individual, every sensor, and various environmental conditions. Consistency of the Bayesian estimators and convergence in distribution of the PDs computed based on the finite element model to those based on the underlying infinite dimensional model are established. Results of human subject data-based numerical studies demonstrating the efficacy of the approach are presented and discussed.

Effects of stomach content on the breath alcohol concentration-transdermal alcohol concentration relationship.

INTRODUCTION: Wearable devices that obtain transdermal alcohol concentration (TAC) could become valuable research tools for monitoring alcohol consumption levels in naturalistic environments if the TAC they produce could be converted into quantitatively-meaningful estimates of breath alcohol concentration (eBrAC). Our team has developed mathematical models to produce eBrAC from TAC, but it is not yet clear how a variety of factors affect the accuracy of the models. Stomach content is one factor that is known to affect breath alcohol concentration (BrAC), but its effect on the BrAC-TAC relationship has not yet been studied. METHODS: We examine the BrAC-TAC relationship by having two investigators participate in four laboratory drinking sessions with varied stomach content conditions: (i) no meal, (ii) half and (iii) full meal before drinking, and (iv) full meal after drinking. BrAC and TAC were obtained every 10 min over the BrAC curve. RESULTS: Eating before drinking lowered BrAC and TAC levels, with greater variability in TAC across person-device pairings, but the BrAC-TAC relationship was not consistently altered by stomach content. The mathematical model calibration parameters, fit indices, and eBrAC curves and summary score outputs did not consistently vary based on stomach content, indicating that our models were able to produce eBrAC from TAC with similar accuracy despite variations in the shape and magnitude of the BrAC curves under different conditions. DISCUSSION AND CONCLUSIONS: This study represents the first examination of how stomach content affects our ability to model estimates of BrAC from TAC and indicates it is not a major factor.

Estimation of the Distribution of Random Parameters in Discrete Time Abstract Parabolic Systems with Unbounded Input and Output: Approximation and Convergence.

A finite dimensional abstract approximation and convergence theory is developed for estimation of the distribution of random parameters in infinite dimensional discrete time linear systems with dynamics described by regularly dissipative operators and involving, in general, unbounded input and output operators. By taking expectations, the system is re-cast as an equivalent abstract parabolic system in a Gelfand triple of Bochner spaces wherein the random parameters become new space-like variables. Estimating their distribution is now analogous to estimating a spatially varying coefficient in a standard deterministic parabolic system. The estimation problems are approximated by a sequence of finite dimensional problems. Convergence is established using a state space-varying version of the Trotter-Kato semigroup approximation theorem. Numerical results for a number of examples involving the estimation of exponential families of densities for random parameters in a diffusion equation with boundary input and output are presented and discussed.

Estimating the Distribution of Random Parameters in a Diffusion Equation Forward Model for a Transdermal Alcohol Biosensor.

We estimate the distribution of random parameters in a distributed parameter model with unbounded input and output for the transdermal transport of ethanol in humans. The model takes the form of a diffusion equation with the input being the blood alcohol concentration and the output being the transdermal alcohol concentration. Our approach is based on the idea of reformulating the underlying dynamical system in such a way that the random parameters are now treated as additional space variables. When the distribution to be estimated is assumed to be defined in terms of a joint density, estimating the distribution is equivalent to estimating the diffusivity in a multi-dimensional diffusion equation and thus well-established finite dimensional approximation schemes, functional analytic based convergence arguments, optimization techniques, and computational methods may all be employed. We use our technique to estimate a bivariate normal distribution based on data for multiple drinking episodes from a single subject.

Comparing a Distributed Parameter Model-Based System Identification Technique with More Conventional Methods for Inverse Problems.

Three methods for the estimation of blood or breath alcohol concentration (BAC/BrAC) from biosensor measured transdermal alcohol concentration (TAC) are evaluated and compared. Specifically, we consider a system identification/quasi-blind deconvolution scheme based on a distributed parameter model with unbounded input and output for ethanol transport in the skin and compare it to two more conventional system identification and filtering/deconvolution techniques for ill-posed inverse problems, one based on frequency domain methods, and the other on a time series approach using an ARMA input/output model. Our basis for comparison are five statistical measures of interest to alcohol researchers and clinicians: peak BAC/BrAC, time of peak BAC/BrAC, the ascending and descending slopes of the BAC/BrAC curve, and the area underneath the BAC/BrAC curve.

Bayesian model selection for the Drosophila gap gene network.

BACKGROUND: The gap gene system controls the early cascade of the segmentation pathway in Drosophila melanogaster as well as other insects. Owing to its tractability and key role in embryo patterning, this system has been the focus for both computational modelers and experimentalists. The gap gene expression dynamics can be considered strictly as a one-dimensional process and modeled as a system of reaction-diffusion equations. While substantial progress has been made in modeling this phenomenon, there still remains a deficit of approaches to evaluate competing hypotheses. Most of the model development has happened in isolation and there has been little attempt to compare candidate models. RESULTS: The Bayesian framework offers a means of doing formal model evaluation. Here, we demonstrate how this framework can be used to compare different models of gene expression. We focus on the Papatsenko-Levine formalism, which exploits a fractional occupancy based approach to incorporate activation of the gap genes by the maternal genes and cross-regulation by the gap genes themselves. The Bayesian approach provides insight about relationship between system parameters. In the regulatory pathway of segmentation, the parameters for number of binding sites and binding affinity have a negative correlation. The model selection analysis supports a stronger binding affinity for Bicoid compared to other regulatory edges, as shown by a larger posterior mean. The procedure doesn't show support for activation of Kruppel by Bicoid. CONCLUSIONS: We provide an efficient solver for the general representation of the Papatsenko-Levine model. We also demonstrate the utility of Bayes factor for evaluating candidate models for spatial pattering models. In addition, by using the parallel tempering sampler, the convergence of Markov chains can be remarkably improved and robust estimates of Bayes factors obtained.

Estimating the quantity and time course of alcohol consumption from transdermal alcohol sensor data: A combined laboratory-ambulatory study.

Transdermal alcohol sensors offer enormous promise for the continuous, objective assessment of alcohol use. Although these sensors have been employed as abstinence monitors for some time now, it is only recently that models have been developed aimed at allowing researchers to derive estimates of the precise amount and time course of drinking, directly from transdermal data. Using data from a combined laboratory-ambulatory study, the current research aims to examine the validity of recently developed methods for estimating BrAC (breath alcohol concentration) directly from transdermal data. Forty-eight heavy social drinkers engaged in 7 days of ambulatory assessment outside the laboratory, and also participated in a laboratory alcohol-administration session. Participants wore the SCRAM transdermal sensor throughout the study, and during the 7 days of ambulatory assessment, they provided daily self-reports of their drinking and also took randomly prompted photographs 6 times per day, which were then evaluated for evidence of alcohol consumption. Results indicated strong associations between daily self-reports of drinking quantity and estimates of BrAC derived from transdermal sensors at both the between- and within-subject level. Data from randomly prompted photos indicated that the time course of estimated BrAC also had validity. Results offer promise for novel methods of estimating BrAC from transdermal data, including those taking a nomothetic (population-based) approach to this estimation, thus potentially adding to our arsenal of techniques for understanding, diagnosing, and ultimately treating alcohol use disorder.

Applying a novel population-based model approach to estimating breath alcohol concentration (BrAC) from transdermal alcohol concentration (TAC) biosensor data.

Alcohol biosensor devices have been developed to unobtrusively measure transdermal alcohol concentration (TAC), the amount of ethanol diffusing through the skin, in nearly continuous fashion in naturalistic settings. Because TAC data are affected by physiological and environmental factors that vary across individuals and drinking episodes, there is not an elementary formula to convert TAC into easily interpretable metrics such as blood and breath alcohol concentrations (BAC/BrAC). In our prior work, we addressed this conversion problem in a deterministic way by developing physics/physiological-based models to convert TAC to estimated BrAC (eBrAC), in which the model parameter values were individually determined for each person wearing a specific transdermal sensor using simultaneously collected TAC (via a biosensor) and BrAC (via a breath analyzer) during a calibration episode. We found these individualized parameter values produced relatively good eBrAC curves for subsequent drinking episodes, but our results also indicated the models were not fully capturing the dynamics of the system and variations across drinking episodes. Here, we report on a novel mathematical framework to improve our ability to model eBrAC from TAC data that uses aggregate population data instead of individualized calibration data to determine model parameter values via a random diffusion equation. We first provide the theoretical mathematical basis for our approach, and then test the efficacy of this method using datasets of contemporaneous BrAC/TAC measurements obtained by a) a single subject during multiple drinking episodes and b) multiple subjects during single drinking episodes. For each dataset, we used a set of drinking episodes to construct the population model, and then ran the model with another set of randomly selected test episodes. We compared raw TAC data to model-simulated TAC curve, breath analyzer BrAC data to model eBrAC curve with 75% credible bands, episode summary scores of peak BrAC, times of peak BrAC, and area under the drinking curve also with 75% credible intervals, and report the percent of the raw BrAC captured within the eBrAC curve credible bands. We also display results when stratifying the data based on the relationship between the raw BrAC and TAC data. Results indicate the population-based model is promising, with better fit within a single participant when stratifying episodes. This study provides initial proof-of-concept for constructing, fitting, and using a population-based model to obtain estimates and error bands for BrAC from TAC. The advancements in this study, including new applications of math, the development of a population-based model with error bars, and the production of corresponding MATLAB codes, represent a major step forward in our ability to produce quantitatively- and temporally-accurate estimates of BrAC from TAC biosensor data.

Time to epileptiform activity and EEG background recovery are independent predictors after cardiac arrest.

OBJECTIVE: Investigate the temporal development of EEG and prognosis. METHODS: Prospective observational substudy of the Target Temperature Management trial. Six sites performed simplified continuous EEG-monitoring (cEEG) on comatose patients after cardiac arrest, blinded to treating physicians. We determined time-points of recovery of a normal-voltage continuous background activity and the appearance of an epileptiform EEG, defined as abundant epileptiform discharges, periodic/rhythmic discharges or electrographic seizure activity. RESULTS: 134 patients were included, 65 had a good outcome. Early recovery of continuous background activity (within 24 h) occurred in 72 patients and predicted good outcome since 55 (76%) had good outcome, increasing the odds for a good outcome seven times compared to a late background recovery. Early appearance of an epileptiform EEG occurred in 38 patients and 34 (89%) had a poor outcome, increasing the odds for a poor outcome six times compared to a late debut. The time to background recovery and the time to epileptiform activity were highly associated with outcome and levels of neuron-specific enolase. Multiple regression analysis showed that both variables were independent predictors. CONCLUSIONS: Time to epileptiform activity and background recovery are independent prognostic indicators. SIGNIFICANCE: Patients with early background recovery combined with late appearance of epileptiform activity may have a good outcome.

A multimodal investigation of contextual effects on alcohol's emotional rewards.

Regular alcohol consumption in unfamiliar social settings has been linked to problematic drinking. A large body of indirect evidence has accumulated to suggest that alcohol's rewarding emotional effects-both negative-mood relieving and positive-mood enhancing-will be magnified when alcohol is consumed within unfamiliar versus familiar social contexts. But empirical research has never directly examined links between contextual familiarity and alcohol reward. In the current study, we mobilized novel ambulatory technology to examine the effect of social familiarity on alcohol reward in everyday drinking contexts while also examining how alcohol reward observed in these field contexts corresponds to reward observed in the laboratory. Heavy social drinking participants (N = 48, 50% male) engaged in an intensive week of ambulatory assessment. Participants wore transdermal alcohol sensors while they reported on their mood and took photographs of their social contexts in response to random prompts. Participants also attended 2 laboratory beverage-administration sessions, during which their emotional responses were assessed and transdermal sensors were calibrated to estimate breathalyzer readings (eBrACs). Results indicated a significant interaction between social familiarity and alcohol episode in everyday drinking settings, with alcohol enhancing mood to a greater extent in relatively unfamiliar versus familiar social contexts. Findings also indicated that drinking in relatively unfamiliar social settings was associated with higher eBrACs. Finally, results indicated a correspondence between some mood effects of alcohol experienced inside and outside the laboratory. This study presents a novel methodology for examining alcohol reward and indicates social familiarity as a promising direction for research seeking to explain problematic drinking. (PsycINFO Database Record

Deconvolving the input to random abstract parabolic systems: a population model-based approach to estimating blood/breath alcohol concentration from transdermal alcohol biosensor data.

The distribution of random parameters in, and the input signal to, a distributed parameter model with unbounded input and output operators for the transdermal transport of ethanol are estimated. The model takes the form of a diffusion equation with the input, which is on the boundary of the domain, being the blood or breath alcohol concentration (BAC/BrAC), and the output, also on the boundary, being the transdermal alcohol concentration (TAC). Our approach is based on the reformulation of the underlying dynamical system in such a way that the random parameters are treated as additional spatial variables. When the distribution to be estimated is assumed to be defined in terms of a joint density, estimating the distribution is equivalent to estimating a functional diffusivity in a multi-dimensional diffusion equation. The resulting system is referred to as a population model, and well-established finite dimensional approximation schemes, functional analytic based convergence arguments, optimization techniques, and computational methods can be used to fit it to population data and to analyze the resulting fit. Once the forward population model has been identified or trained based on a sample from the population, the resulting distribution can then be used to deconvolve the BAC/BrAC input signal from the biosensor observed TAC output signal formulated as either a quadratic programming or linear quadratic tracking problem. In addition, our approach allows for the direct computation of corresponding credible bands without simulation. We use our technique to estimate bivariate normal distributions and deconvolve BAC/BrAC from TAC based on data from a population that consists of multiple drinking episodes from a single subject and a population consisting of single drinking episodes from multiple subjects.

Obtaining continuous BrAC/BAC estimates in the field: A hybrid system integrating transdermal alcohol biosensor, Intellidrink smartphone app, and BrAC Estimator software tools.

Biosensors have been developed to measure transdermal alcohol concentration (TAC), but converting TAC into interpretable indices of blood/breath alcohol concentration (BAC/BrAC) is difficult because of variations that occur in TAC across individuals, drinking episodes, and devices. We have developed mathematical models and the BrAC Estimator software for calibrating and inverting TAC into quantifiable BrAC estimates (eBrAC). The calibration protocol to determine the individualized parameters for a specific individual wearing a specific device requires a drinking session in which BrAC and TAC measurements are obtained simultaneously. This calibration protocol was originally conducted in the laboratory with breath analyzers used to produce the BrAC data. Here we develop and test an alternative calibration protocol using drinking diary data collected in the field with the smartphone app Intellidrink to produce the BrAC calibration data. We compared BrAC Estimator software results for 11 drinking episodes collected by an expert user when using Intellidrink versus breath analyzer measurements as BrAC calibration data. Inversion phase results indicated the Intellidrink calibration protocol produced similar eBrAC curves and captured peak eBrAC to within 0.0003%, time of peak eBrAC to within 18min, and area under the eBrAC curve to within 0.025% alcohol-hours as the breath analyzer calibration protocol. This study provides evidence that drinking diary data can be used in place of breath analyzer data in the BrAC Estimator software calibration procedure, which can reduce participant and researcher burden and expand the potential software user pool beyond researchers studying participants who can drink in the laboratory.

THE PROHOROV METRIC FRAMEWORK AND AGGREGATE DATA INVERSE PROBLEMS FOR RANDOM PDEs.

We consider nonparametric estimation of probability measures for parameters in problems where only aggregate (population level) data are available. We summarize an existing computational method for the estimation problem which has been developed over the past several decades [24, 5, 12, 28, 16]. Theoretical results are presented which establish the existence and consistency of very general (ordinary, generalized and other) least squares estimates and estimators for the measure estimation problem with specific application to random PDEs.