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A patient's digital twin is a computational model that describes the evolution of their health over time. Digital twins have the potential to revolutionize medicine by enabling individual-level computer simulations of human health, which can be used to conduct more efficient clinical trials or to recommend personalized treatment options. Due to the overwhelming complexity of human biology, machine learning approaches that leverage large datasets of historical patients' longitudinal health records to generate patients' digital twins are more tractable than potential mechanistic models. In this manuscript, we describe a neural network architecture that can learn conditional generative models of clinical trajectories, which we call Digital Twin Generators (DTGs), that can create digital twins of individual patients. We show that the same neural network architecture can be trained to generate accurate digital twins for patients across 13 different indications simply by changing the training set and tuning hyperparameters. By introducing a general purpose architecture, we aim to unlock the ability to scale machine learning approaches to larger datasets and across more indications so that a digital twin could be created for any patient in the world.

Randomized controlled trials (RCTs) with binary primary endpoints introduce novel challenges for inferring the causal effects of treatments. The most significant challenge is non-collapsibility, in which the conditional odds ratio estimand under covariate adjustment differs from the unconditional estimand in the logistic regression analysis of RCT data. This issue gives rise to apparent paradoxes, such as the variance of the estimator for the conditional odds ratio from a covariate-adjusted model being greater than the variance of the estimator from the unadjusted model. We address this challenge in the context of adjustment based on predictions of control outcomes from generative artificial intelligence (AI) algorithms, which are referred to as prognostic scores. We demonstrate that prognostic score adjustment in logistic regression increases the power of the Wald test for the conditional odds ratio under a fixed sample size, or alternatively reduces the necessary sample size to achieve a desired power, compared to the unadjusted analysis. We derive formulae for prospective calculations of the power gain and sample size reduction that can result from adjustment for the prognostic score. Furthermore, we utilize g-computation to expand the scope of prognostic score adjustment to inferences on the marginal risk difference, relative risk, and odds ratio estimands. We demonstrate the validity of our formulae via extensive simulation studies that encompass different types of logistic regression model specifications. Our simulation studies also indicate how prognostic score adjustment can reduce the variance of g-computation estimators for the marginal estimands while maintaining frequentist properties such as asymptotic unbiasedness and Type I error rate control. Our methodology can ultimately enable more definitive and conclusive analyses for RCTs with binary primary endpoints., Comment: 27 pages, 1 figure, 9 tables

Effective and rapid decision-making from randomized controlled trials (RCTs) requires unbiased and precise treatment effect inferences. Two strategies to address this requirement are to adjust for covariates that are highly correlated with the outcome, and to leverage historical control information via Bayes' theorem. We propose a new Bayesian prognostic covariate adjustment methodology, referred to as Bayesian PROCOVA, that combines these two strategies. Covariate adjustment in Bayesian PROCOVA is based on generative artificial intelligence (AI) algorithms that construct a digital twin generator (DTG) for RCT participants. The DTG is trained on historical control data and yields a digital twin (DT) probability distribution for each RCT participant's outcome under the control treatment. The expectation of the DT distribution, referred to as the prognostic score, defines the covariate for adjustment. Historical control information is leveraged via an additive mixture prior with two components: an informative prior probability distribution specified based on historical control data, and a weakly informative prior distribution. The mixture weight determines the extent to which posterior inferences are drawn from the informative component, versus the weakly informative component. This weight has a prior distribution as well, and so the entire additive mixture prior is completely pre-specifiable without involving any RCT information. We establish an efficient Gibbs algorithm for sampling from the posterior distribution, and derive closed-form expressions for the posterior mean and variance of the treatment effect parameter conditional on the weight, in Bayesian PROCOVA. We evaluate efficiency gains of Bayesian PROCOVA via its bias control and variance reduction compared to frequentist PROCOVA in simulation studies that encompass different discrepancies. These gains translate to smaller RCTs., Comment: 33 pages, 12 figures, 2 tables; Added a new appendix section (conclusions unchanged)

A crucial task for a randomized controlled trial (RCT) is to specify a statistical method that can yield an efficient estimator and powerful test for the treatment effect. A novel and effective strategy to obtain efficient and powerful treatment effect inferences is to incorporate predictions from generative artificial intelligence (AI) algorithms into covariate adjustment for the regression analysis of a RCT. Training a generative AI algorithm on historical control data enables one to construct a digital twin generator (DTG) for RCT participants, which utilizes a participant's baseline covariates to generate a probability distribution for their potential control outcome. Summaries of the probability distribution from the DTG are highly predictive of the trial outcome, and adjusting for these features via regression can thus improve the quality of treatment effect inferences, while satisfying regulatory guidelines on statistical analyses, for a RCT. However, a critical assumption in this strategy is homoskedasticity, or constant variance of the outcome conditional on the covariates. In the case of heteroskedasticity, existing covariate adjustment methods yield inefficient estimators and underpowered tests. We propose to address heteroskedasticity via a weighted prognostic covariate adjustment methodology (Weighted PROCOVA) that adjusts for both the mean and variance of the regression model using information obtained from the DTG. We prove that our method yields unbiased treatment effect estimators, and demonstrate via comprehensive simulation studies and case studies from Alzheimer's disease that it can reduce the variance of the treatment effect estimator, maintain the Type I error rate, and increase the power of the test for the treatment effect from 80% to 85%~90% when the variances from the DTG can explain 5%~10% of the variation in the RCT participants' outcomes., Comment: 49 pages, 6 figures, 12 tables

Randomized Controlled Trials (RCTs) often adjust for baseline covariates in order to increase power. This technical note provides a short derivation of a simple rule of thumb for approximating the ratio of the power of an adjusted analysis to that of an unadjusted analysis. Specifically, if the unadjusted analysis is powered to approximately 80\%, then the ratio of the power of the adjusted analysis to the power of the unadjusted analysis is approximately $1 + \frac{1}{2} R^2$, where $R$ is the correlation between the baseline covariate and the outcome.

Conditional generative models are capable of using contextual information as input to create new imaginative outputs. Conditional Restricted Boltzmann Machines (CRBMs) are one class of conditional generative models that have proven to be especially adept at modeling noisy discrete or continuous data, but the lack of expressivity in CRBMs have limited their widespread adoption. Here we introduce Neural Boltzmann Machines (NBMs) which generalize CRBMs by converting each of the CRBM parameters to their own neural networks that are allowed to be functions of the conditional inputs. NBMs are highly flexible conditional generative models that can be trained via stochastic gradient descent to approximately maximize the log-likelihood of the data. We demonstrate the utility of NBMs especially with normally distributed data which has historically caused problems for Gaussian-Bernoulli CRBMs. Code to reproduce our results can be found at https://github.com/unlearnai/neural-boltzmann-machines., Comment: 7 pages, 4 figures

Restricted Boltzmann Machines (RBMs) are probabilistic generative models that can be trained by maximum likelihood in principle, but are usually trained by an approximate algorithm called Contrastive Divergence (CD) in practice. In general, a CD-k algorithm estimates an average with respect to the model distribution using a sample obtained from a k-step Markov Chain Monte Carlo Algorithm (e.g., block Gibbs sampling) starting from some initial configuration. Choices of k typically vary from 1 to 100. This technical report explores if it's possible to leverage a simple approximate sampling algorithm with a modified version of CD in order to train an RBM with k=0. As usual, the method is illustrated on MNIST.

Alzheimer's Disease (AD) is a neurodegenerative disease that affects subjects in a broad range of severity and is assessed in clinical trials with multiple cognitive and functional instruments. As clinical trials in AD increasingly focus on earlier stages of the disease, especially Mild Cognitive Impairment (MCI), the ability to model subject outcomes across the disease spectrum is extremely important. We use unsupervised machine learning models called Conditional Restricted Boltzmann Machines (CRBMs) to create Digital Twins of AD subjects. Digital Twins are simulated clinical records that share baseline data with actual subjects and comprehensively model their outcomes under standard-of-care. The CRBMs are trained on a large set of records from subjects in observational studies and the placebo arms of clinical trials across the AD spectrum. These data exhibit a challenging, but common, patchwork of measured and missing observations across subjects in the dataset, and we present a novel model architecture designed to learn effectively from it. We evaluate performance against a held-out test dataset and show how Digital Twins simultaneously capture the progression of a number of key endpoints in clinical trials across a broad spectrum of disease severity, including MCI and mild-to-moderate AD.

Historical data about disease outcomes can be integrated into the analysis of clinical trials in many ways. We build on existing literature that uses prognostic scores from a predictive model to increase the efficiency of treatment effect estimates via covariate adjustment. Here we go further, utilizing a Bayesian framework that combines prognostic covariate adjustment with an empirical prior distribution learned from the predictive performances of the prognostic model on past trials. The Bayesian approach interpolates between prognostic covariate adjustment with strict type I error control when the prior is diffuse, and a single-arm trial when the prior is sharply peaked. This method is shown theoretically to offer a substantial increase in statistical power, while limiting the type I error rate under reasonable conditions. We demonstrate the utility of our method in simulations and with an analysis of a past Alzheimer's disease clinical trial., Comment: 28 pages, 11 figures

Estimating causal effects from randomized experiments is central to clinical research. Reducing the statistical uncertainty in these analyses is an important objective for statisticians. Registries, prior trials, and health records constitute a growing compendium of historical data on patients under standard-of-care that may be exploitable to this end. However, most methods for historical borrowing achieve reductions in variance by sacrificing strict type-I error rate control. Here, we propose a use of historical data that exploits linear covariate adjustment to improve the efficiency of trial analyses without incurring bias. Specifically, we train a prognostic model on the historical data, then estimate the treatment effect using a linear regression while adjusting for the trial subjects' predicted outcomes (their prognostic scores). We prove that, under certain conditions, this prognostic covariate adjustment procedure attains the minimum variance possible among a large class of estimators. When those conditions are not met, prognostic covariate adjustment is still more efficient than raw covariate adjustment and the gain in efficiency is proportional to a measure of the predictive accuracy of the prognostic model above and beyond the linear relationship with the raw covariates. We demonstrate the approach using simulations and a reanalysis of an Alzheimer's Disease clinical trial and observe meaningful reductions in mean-squared error and the estimated variance. Lastly, we provide a simplified formula for asymptotic variance that enables power calculations that account for these gains. Sample size reductions between 10% and 30% are attainable when using prognostic models that explain a clinically realistic percentage of the outcome variance.

Multiple Sclerosis (MS) is a neurodegenerative disorder characterized by a complex set of clinical assessments. We use an unsupervised machine learning model called a Conditional Restricted Boltzmann Machine (CRBM) to learn the relationships between covariates commonly used to characterize subjects and their disease progression in MS clinical trials. A CRBM is capable of generating digital twins, which are simulated subjects having the same baseline data as actual subjects. Digital twins allow for subject-level statistical analyses of disease progression. The CRBM is trained using data from 2395 subjects enrolled in the placebo arms of clinical trials across the three primary subtypes of MS. We discuss how CRBMs are trained and show that digital twins generated by the model are statistically indistinguishable from their actual subject counterparts along a number of measures.

Most approaches to machine learning from electronic health data can only predict a single endpoint. Here, we present an alternative that uses unsupervised deep learning to simulate detailed patient trajectories. We use data comprising 18-month trajectories of 44 clinical variables from 1908 patients with Mild Cognitive Impairment or Alzheimer's Disease to train a model for personalized forecasting of disease progression. We simulate synthetic patient data including the evolution of each sub-component of cognitive exams, laboratory tests, and their associations with baseline clinical characteristics, generating both predictions and their confidence intervals. Our unsupervised model predicts changes in total ADAS-Cog scores with the same accuracy as specifically trained supervised models and identifies sub-components associated with word recall as predictive of progression. The ability to simultaneously simulate dozens of patient characteristics is a crucial step towards personalized medicine for Alzheimer's Disease.

Restricted Boltzmann Machines (RBMs) are a class of generative neural network that are typically trained to maximize a log-likelihood objective function. We argue that likelihood-based training strategies may fail because the objective does not sufficiently penalize models that place a high probability in regions where the training data distribution has low probability. To overcome this problem, we introduce Boltzmann Encoded Adversarial Machines (BEAMs). A BEAM is an RBM trained against an adversary that uses the hidden layer activations of the RBM to discriminate between the training data and the probability distribution generated by the model. We present experiments demonstrating that BEAMs outperform RBMs and GANs on multiple benchmarks.

Machine Learning (ML) is one of the most exciting and dynamic areas of modern research and application. The purpose of this review is to provide an introduction to the core concepts and tools of machine learning in a manner easily understood and intuitive to physicists. The review begins by covering fundamental concepts in ML and modern statistics such as the bias-variance tradeoff, overfitting, regularization, generalization, and gradient descent before moving on to more advanced topics in both supervised and unsupervised learning. Topics covered in the review include ensemble models, deep learning and neural networks, clustering and data visualization, energy-based models (including MaxEnt models and Restricted Boltzmann Machines), and variational methods. Throughout, we emphasize the many natural connections between ML and statistical physics. A notable aspect of the review is the use of Python Jupyter notebooks to introduce modern ML/statistical packages to readers using physics-inspired datasets (the Ising Model and Monte-Carlo simulations of supersymmetric decays of proton-proton collisions). We conclude with an extended outlook discussing possible uses of machine learning for furthering our understanding of the physical world as well as open problems in ML where physicists may be able to contribute. (Notebooks are available at https://physics.bu.edu/~pankajm/MLnotebooks.html ), Comment: Notebooks have been updated. 122 pages, 78 figures, 20 Python notebooks