Breimans classic paper casts data analysis as a choice between two cultures: data modelers and algorithmic modelers. Stated broadly, data modelers use simple, interpretable models with well-understood theoretical properties to analyze data. Algorithm
ic modelers prioritize predictive accuracy and use more flexible function approximations to analyze data. This dichotomy overlooks a third set of models $-$ mechanistic models derived from scientific theories (e.g., ODE/SDE simulators). Mechanistic models encode application-specific scientific knowledge about the data. And while these categories represent extreme points in model space, modern computational and algorithmic tools enable us to interpolate between these points, producing flexible, interpretable, and scientifically-informed hybrids that can enjoy accurate and robust predictions, and resolve issues with data analysis that Breiman describes, such as the Rashomon effect and Occams dilemma. Challenges still remain in finding an appropriate point in model space, with many choices on how to compose model components and the degree to which each component informs inferences.
We develop a new model of insulin-glucose dynamics for forecasting blood glucose in type 1 diabetics. We augment an existing biomedical model by introducing time-varying dynamics driven by a machine learning sequence model. Our model maintains a phys
iologically plausible inductive bias and clinically interpretable parameters -- e.g., insulin sensitivity -- while inheriting the flexibility of modern pattern recognition algorithms. Critical to modeling success are the flexible, but structured representations of subject variability with a sequence model. In contrast, less constrained models like the LSTM fail to provide reliable or physiologically plausible forecasts. We conduct an extensive empirical study. We show that allowing biomedical model dynamics to vary in time improves forecasting at long time horizons, up to six hours, and produces forecasts consistent with the physiological effects of insulin and carbohydrates.
Stochastic gradient MCMC (SG-MCMC) algorithms have proven useful in scaling Bayesian inference to large datasets under an assumption of i.i.d data. We instead develop an SG-MCMC algorithm to learn the parameters of hidden Markov models (HMMs) for tim
e-dependent data. There are two challenges to applying SG-MCMC in this setting: The latent discrete states, and needing to break dependencies when considering minibatches. We consider a marginal likelihood representation of the HMM and propose an algorithm that harnesses the inherent memory decay of the process. We demonstrate the effectiveness of our algorithm on synthetic experiments and an ion channel recording data, with runtimes significantly outperforming batch MCMC.
Dependent nonparametric processes extend distributions over measures, such as the Dirichlet process and the beta process, to give distributions over collections of measures, typically indexed by values in some covariate space. Such models are appropr
iate priors when exchangeability assumptions do not hold, and instead we want our model to vary fluidly with some set of covariates. Since the concept of dependent nonparametric processes was formalized by MacEachern [1], there have been a number of models proposed and used in the statistics and machine learning literatures. Many of these models exhibit underlying similarities, an understanding of which, we hope, will help in selecting an appropriate prior, developing new models, and leveraging inference techniques.
