Gaussian Process Regression

Concept

Theoretically, Gaussian Process Regression is a non-parametric Bayesian regression method that models an unknown function as a Gaussian process, allowing predictions to be made while explicitly quantifying uncertainty. Rather than assuming a fixed functional form, the method defines a probability distribution over possible functions, with relationships between observations determined by a covariance (kernel) function. In health economics, Gaussian Process Regression is widely used for surrogate modelling (emulation), probabilistic sensitivity analysis, value of information analysis and optimisation of computationally expensive decision models.

Mathematically, Gaussian Process Regression assumes that function values jointly follow a multivariate normal distribution specified by a mean function and covariance kernel. Predictions for new observations are obtained by conditioning the joint Gaussian distribution on observed data, producing both posterior mean estimates and posterior predictive variances. The mathematical framework enables interpolation of complex, nonlinear relationships while providing a measure of prediction uncertainty.

In practice, Gaussian Process Regression is fitted using simulation outputs generated from health economic models. Hyperparameters governing the covariance function are estimated by maximising the marginal likelihood or through Bayesian inference. The resulting emulator provides rapid approximations to computationally intensive models and is commonly applied in expected value of perfect information (EVPI), expected value of partial perfect information (EVPPI), probabilistic sensitivity analysis and model calibration.

---

Purpose

Used to construct computationally efficient surrogate models that approximate complex health economic models while quantifying prediction uncertainty for decision analysis, value of information analysis and model optimisation.

---

Mathematical Formulae

Primary Formula

Prior Gaussian process:

f(x) ? GP(m(x), k(x, x?))

Posterior predictive mean:

? = K?(K + �?�I)??y**

where:

• m(x) = mean function
• k(x, x?) = covariance (kernel) function
• K = covariance matrix of observed inputs
• K* = covariance between observed and prediction points
• �?� = observation noise variance
• y = observed responses

Supporting Formulae

Posterior predictive variance:

�� = k(x, x) ? K?(K + �?�I)??K***

Common squared exponential kernel:

k(x, x?) = �?� exp(??x ? x??�/(2l�))

where:

• �?� = signal variance
• l = length-scale parameter

Related Mathematical Methods

• Bayesian regression
• Gaussian processes
• Kernel methods
• Emulator modelling
• Maximum likelihood estimation
• Bayesian optimisation
• Expected Value of Perfect Information (EVPI)
• Expected Value of Partial Perfect Information (EVPPI)
• Probabilistic sensitivity analysis

---

Example

A probabilistic Markov model evaluating a vaccination programme requires six minutes to complete one simulation.

Five hundred model evaluations are generated using different combinations of vaccine effectiveness, treatment costs and disease incidence. A Gaussian Process Regression emulator is fitted to these outputs. The emulator predicts incremental net monetary benefit for thousands of additional parameter combinations within seconds while providing posterior predictive uncertainty. The emulator is subsequently used to estimate EVPPI without repeatedly executing the original decision model.

---

Excel Implementation

Function	Example Formula	Health Economics Application
LINEST	=LINEST(Y2:Y101,X2:X101,TRUE,TRUE)	Perform simple regression approximations during preliminary model exploration.
MMULT	=MMULT(A2:F7,G2:G7)	Matrix multiplication used when implementing covariance calculations.
MINVERSE	=MINVERSE(A2:F7)	Invert covariance matrices in simplified Gaussian Process implementations.
TRANSPOSE	=TRANSPOSE(A2:F7)	Construct covariance matrix operations.

---

VBA (Optional)

Automate the generation of simulation datasets, covariance matrix calculations and emulator predictions for large-scale probabilistic sensitivity and value of information analyses.

---

Sources

• Rasmussen CE, Williams CKI. Gaussian Processes for Machine Learning. MIT Press.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• Strong M, Oakley JE, Brennan A. Estimating Multiparameter Partial Expected Value of Perfect Information from a Probabilistic Sensitivity Analysis Sample: A Nonparametric Regression Approach. Medical Decision Making. 2014.
• Heath A, Manolopoulou I, Baio G. Efficient Monte Carlo Estimation of the Expected Value of Sample Information Using Gaussian Process Regression. Medical Decision Making. 2018.
• NICE. Health Technology Evaluation Manual.
• ISPOR-SMDM Modeling Good Research Practices Task Force Reports.