Bayesian Optimisation

Concept

Theoretically, Bayesian Optimisation is a sequential optimisation method for identifying the optimum of an objective function that is expensive to evaluate, stochastic or lacks an analytical form. It combines Bayesian statistical inference with decision theory to iteratively update beliefs about the objective function and determine where subsequent evaluations should be performed. In health economics, Bayesian optimisation is used to calibrate complex decision models, estimate model parameters and optimise resource allocation when repeated model evaluations are computationally intensive.

Mathematically, Bayesian optimisation represents the unknown objective function using a probabilistic surrogate model, most commonly a Gaussian process. After each function evaluation, Bayes' theorem is used to update the posterior distribution of the objective function. An acquisition function balances exploration of uncertain regions with exploitation of promising solutions by selecting the next evaluation point that maximises the expected improvement or another optimisation criterion.

In practice, Bayesian optimisation is implemented by defining an objective function, specifying prior distributions for the surrogate model, selecting an acquisition function and sequentially updating the posterior as additional model evaluations are performed. It is applied in health economic model calibration, parameter estimation, simulation optimisation, machine learning and computational decision modelling where conventional optimisation methods would require excessive computational time.

---

Purpose

Used to efficiently optimise computationally expensive health economic models by sequentially selecting parameter values that maximise model performance while minimising the number of model evaluations.

---

Mathematical Formulae

Primary Formula

Bayesian posterior distribution:

p(f ? D) = [p(D ? f) p(f)] / p(D)

where:

• f = unknown objective function
• D = observed evaluations
• p(f) = prior distribution
• p(D ? f) = likelihood
• p(f ? D) = posterior distribution

Supporting Formulae

Expected Improvement acquisition function:

EI(x) = E[max(f(x) ? f(x?), 0)]

where:

• x = candidate evaluation point
• f(x?) = current best objective value

Related Mathematical Methods

• Gaussian process regression
• Bayesian inference
• Sequential optimisation
• Global optimisation
• Surrogate modelling
• Expected Improvement
• Upper Confidence Bound optimisation

---

Example

A Markov model contains four uncertain transition probabilities requiring calibration to observed survival data. Each model evaluation requires 30 minutes of computation.

Bayesian optimisation begins with 20 initial parameter sets, constructs a Gaussian process surrogate model, and uses the Expected Improvement criterion to select subsequent parameter combinations. After 65 total evaluations, the optimal parameter set is identified, whereas an exhaustive grid search would have required several thousand model evaluations.

---

Excel Implementation

Function	Example Formula	Health Economics Application
Solver	Solver Optimisation	Performs local optimisation once candidate solutions have been identified.
NORM.DIST	=NORM.DIST(B2,Mean,SD,TRUE)	Evaluates normal distributions used in Gaussian process approximations.
INDEX	=INDEX(B2:B100,MATCH(MAX(C2:C100),C2:C100,0))	Retrieves the best-performing parameter set.
MAX	=MAX(C2:C100)	Identifies the current optimum objective value during sequential optimisation.

---

VBA (Optional)

Automate iterative parameter evaluation by linking Excel models with external Bayesian optimisation routines and recording successive posterior updates.

---

Sources

• Rasmussen CE, Williams CKI. Gaussian Processes for Machine Learning. MIT Press.
• Brochu E, Cora VM, de Freitas N. A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modelling and Hierarchical Reinforcement Learning. University of British Columbia.
• Snoek J, Larochelle H, Adams RP. Practical Bayesian Optimization of Machine Learning Algorithms. Advances in Neural Information Processing Systems.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
• ISPOR-SMDM Modeling Good Research Practices Task Force reports.