Data-Informed Modeling in the Health Sciences

Abstract

The adoption of automation and technology by health professionals is triggering an explosion of databases and data streams in that sector. The emergence of this data torrent creates the pressing need to mine it for value, which in turn requires investment for the development of modeling and analysis tools. In view of this, dynamicists are presented with the terrific opportunity to enrich their discipline by supplying it with new tools, expanding its scope, and elevating its social impact. This chapter is written in that spirit, examining three concrete case studies encountered in the field: quantifying the salmonellosis risk posed by distinct food sources, assimilating genetic data into a dynamical model for avian influenza transmission, and statistically decontaminating gas chromatography/mass spectroscopy time series. We review available prototypical models and build on them guided by data and mathematical abstraction, demonstrating in the process how to root a model into data. This takes us quite naturally into the realm of probabilistic and statistical modeling and reopens a decades-old discussion on the role of discrete models in applied mathematics. We also touch briefly on the timely subject of mathematicians being employed as such outside math departments and attempt a short outlook on their prospects and opportunities.

Appendix: A Short Primer on Parameter Estimation

The fundamental belief underpinning any modeling endeavor is that system measurements can be approximately generated by a specific model. In general terms, inference uses such measurements to mitigate uncertainty present in the underlying model. In this short appendix, we assume a well-defined class of candidate models that differ only in particulars; our task is to locate among them the one that best fits the available measurements (data). Here, these models share a common functional form containing finitely many parameters, so we speak of a parametric family and parametric inference. Lifting the uncertainty surrounding the parameter values is the inferential task par excellence.

Parameter values can be inferred in various ways joined by a common thread. Typically, unknown values are obtained as solutions to an optimization problem involving the model class and available data; in the problems treated here, that data is model outputs such as values of the dependent variables. For a deterministic model, a reasonable minimal requirement for an estimator would seemingly be self-consistency: given data generated by simulating a model with specific parameter values, a self-consistent estimator would return those precise parameter values, i.e., invert the simulation. Imposing that condition is reasonable, as long as distinct parameter values yield well-defined, distinct data (parameter identifiability [24]). However, the models treated in this chapter are probabilistic: specific parameter settings only have a certain probability to generate specific data. This makes the correspondence between parameter values and data both one-to-many and many-to-one, and it necessitates rethinking what can be reasonably expected from an estimator.

To address this problem, we start with univariate r.v.s X ₁, …, X _N defined on a common sample space Ω and having distributions $f_{X_1},\ldots ,f_{X_N}$. We then write $X = (X_1,\ldots ,X_N) : \varOmega \to \mathcal {X}$ for the multivariate r.v. collecting them, and we recognize as the space where data resides. This data space is equipped with an induced joint probability distribution , and each point x = (x ₁, …, x _N) in it corresponds to a full set of system measurements. In general, this joint distribution does not follow trivially from the marginals $f_{X_1},\ldots ,f_{X_N}$; determining it may be a sizable part of the modeling process and a closed-form expression outside reach, if the problem does not possess additional structure. A favorable case occurs when X ₁, …, X _N are pairwise independent, as f _X then has the product decomposition $f_X(x) = \prod _{n=1}^N f_{X_n}(x_n)$; another, trivial case occurs when r.v. components are algebraically constrained. Often, neither is true and modeling f _X is nontrivial. As a concrete example, the reader should derive the sampling distribution of $X = \sum _{n=1}^N X_n/N$ (sample mean) corresponding to i.i.d. Gaussian r.v.s X ₁, …, X _N.

We now assume that f _X depends on a set of parameters Θ = (θ ₁, …, θ _M) ∈ Δ and write f _X|Θ(⋅|θ) to reflect this. The parameter values θ are the subject of inference, i.e., of mapping data to parameter values by means of an estimator $\hat {\varTheta } : \mathcal {X} \to \varDelta$. This function will unambiguously (i.e., deterministically) map specific data to specific parameter values without recourse to the parameter values that generated the data. It is in this sense that parameter estimation reverse-engineers data generation. To proceed intelligently with estimator design, we note that parameter values generate data probabilistically—by sampling f _X|Θ(⋅|θ)—but $\hat {\varTheta }$ maps these to parameter estimates deterministically. The combination of sampling and estimation is therefore probabilistic in nature, meaning that a fixed set of parameter values generates different data and thus gives rise to various estimates of those values. In fact, the composite map $\hat {\varTheta } \circ X : \varOmega \to \varDelta$ is a transformed version of X and hence automatically an r.v. in its own right. Indeed, any measurable set U in parameter space Δ is assigned the measure of its pre-image $\hat {\varTheta }^{-1}(U)$ in data space $\mathcal {X}$ which, in turn, inherits that of $X^{-1}(\hat {\varTheta }^{-1}(U))$ in sample space Ω.

Being a r.v., the estimator is distributed according to some sampling distribution $f_{\hat {\varTheta }\vert \varTheta }$ that depends on the unknown parameters values. This observation suggests adapting the deterministic notion of self-consistency to that of an unbiased estimator, which amounts to demanding that

$$\displaystyle \begin{aligned} \int_{\varDelta} \hat{\theta} \, f_{\hat{\varTheta}\vert\varTheta}(\hat{\theta}\vert\theta) \, \mathrm{d}\hat{\theta} \ = \ \int_{\mathcal{X}} \hat{\varTheta}(x) \, f_{X\vert\varTheta}(x\vert\theta) \, \mathrm{d}x \ = \ \theta , \quad \theta\in\varDelta . \end{aligned} $$

(6.4.11)

If this condition holds, then the expected parameter estimates match the true parameter values, i.e., the estimator is correct on average although individual estimates inevitably deviate from the truth. That deviation can be quantified (again on average) using the variance of $f_{\hat {\varTheta }\vert \varTheta }$, which one would like to keep as low as possible; note that some variance is inevitable, see the Cramér–Rao bound [11]. These notions of estimator bias and variance permeate estimation theory fundamentally. For example, the aforementioned variance bound links to information theory and geometry [1], whereas modern machine learning work often involves biased estimators that trade off accuracy for precision.

In our work in this chapter, we employed the likelihood L(θ|x) = f _X|Θ(x|θ) with which parameter values θ ∈ Δ generate given data $x\in \mathcal {X}$. We specifically used the maximum likelihood estimator (MLE),

$$\displaystyle \begin{aligned} \hat{\varTheta}(x) = \mathrm{arg} \, \max_\theta L(\theta \vert x) = \mathrm{arg} \, \max_\theta f_{X\vert\varTheta}(x\vert\theta) , \quad x\in\mathcal{X} . {} \end{aligned} $$

(6.4.12)

In words, the estimate for the parameter value generating given data is the value maximizing the probability (likelihood) of generating that data. The evident circularity in this statement manifests that sampling and inference run contrary to each other. Note that neither existence nor uniqueness of the MLE is automatic (nor universal) and that the MLE is often biased. However, if X ₁, …, X _N are i.i.d. and N →∞, then $f_{\hat {\varTheta }\vert \varTheta }(\cdot \vert \theta )$ is an approximate Gaussian centered at θ by the central limit theorem (CLT). For more detailed introductions to parameter inference at two different levels, we refer the reader to [10, 25].