Gaussian Process

Concept

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Theoretically, a Gaussian Process is a stochastic process in which every finite collection of function values follows a multivariate normal distribution. It provides a Bayesian non-parametric framework for representing uncertainty about an unknown function through a mean function and a covariance function. In health economics, Gaussian processes are used as flexible surrogate models for computationally intensive decision models, model calibration and value of information analysis.

Mathematically, a Gaussian process defines a probability distribution over functions and is written as f(x) ? GP(m(x), k(x, x?)), where m(x) is the mean function and k(x, x?) is the covariance kernel. Predictions at unobserved inputs are obtained from the conditional multivariate normal distribution determined by the observed data, the selected covariance function and any assumed observation noise.

In practice, Gaussian processes are estimated by selecting a covariance kernel, estimating its hyperparameters and conditioning the process on observed model outputs. Health economists use Gaussian process regression to emulate complex simulation models, estimate Expected Value of Partial Perfect Information and accelerate calibration, optimisation and uncertainty analysis when direct model evaluation is computationally expensive.

Purpose

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Used to model uncertain nonlinear relationships, emulate computationally intensive health economic models and provide probabilistic predictions with quantified uncertainty.

Mathematical Formulae

Primary Formula

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

where:

f(x) = unknown function evaluated at input x

m(x) = mean function

k(x, x?) = covariance function between inputs x and x?

Supporting Formulae

Mean function:

m(x) = E[f(x)]

Covariance function:

k(x, x?) = Cov(f(x), f(x?))

Predictive mean:

?* = K*?(K + ��I)??y

Predictive covariance:

�* = K** ? K*?(K + ��I)??K*

where:

K = covariance matrix for observed inputs

K* = covariance between observed and prediction inputs

K** = covariance matrix for prediction inputs

�� = observation noise variance

I = identity matrix

y = observed outputs

Related Mathematical Methods

• Bayesian Regression
• Multivariate Normal Distribution
• Covariance Matrix
• Kernel Method
• Gaussian Process Regression
• Surrogate Modelling
• Bayesian Optimisation
• Expected Value of Partial Perfect Information

Example

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A health economic microsimulation requires several hours to evaluate one combination of uncertain parameters. A Gaussian process is fitted to 500 completed model evaluations using treatment effect, disease progression and cost parameters as inputs and incremental net monetary benefit as the output.

The fitted Gaussian process predicts incremental net monetary benefit for untested parameter combinations and provides a predictive variance for each estimate. These predictions are then used to approximate the model across the parameter space and estimate EVPPI without repeatedly running the full microsimulation.

Excel Implementation

Function	Example Formula	Health Economics Application
AVERAGE	=AVERAGE(B2:B501)	Estimate a constant mean function from observed model outputs.
COVARIANCE.S	=COVARIANCE.S(B2:B501,C2:C501)	Estimate empirical covariance between model inputs or outputs.
MMULT	=MMULT(B2:F2,G2:G6)	Perform matrix multiplication used in Gaussian process prediction calculations.
MINVERSE	=MINVERSE(B2:F6)	Calculate the inverse of a covariance matrix for small Gaussian process applications.
EXP	=EXP(-((A2-B2)^2)/(2*$H$2^2))	Calculate a squared exponential covariance value between two inputs.

VBA (Optional)

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VBA can automate covariance matrix construction and Gaussian process prediction for small surrogate modelling applications.

Sources

• Rasmussen CE, Williams CKI. Gaussian Processes for Machine Learning. MIT Press.
• Oakley JE, O?Hagan A. Probabilistic sensitivity analysis of complex models: a Bayesian approach. Journal of the Royal Statistical Society: Series B. 2004;66(3):751?769.
• Strong M, Oakley JE, Brennan A. Estimating multiparameter partial expected value of perfect information from a probabilistic sensitivity analysis sample. Medical Decision Making. 2014;34(3):311?326.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• ISPOR Value of Information Good Practice Reports.