The Concept of MLGLUE#
In a nutshell, MLGLUE can be used to accelerate Bayesian inference of (hydrological) model parameters. We assume that you are familiar with the general concept of Bayesian inference. For more details, references, etc., please also look at the original publication about MLGLUE here.
Inverse problems and Bayesian inference#
We consider a model \(\mathcal{F}(\cdot)\), which simulates observed values \(\mathbf{d}\) (up to errors \(\boldsymbol \varepsilon\)), using model parameters \(\boldsymbol \theta\):
Our aim when solving an inverse problem is to find parameters of the model such that the model simulations match the corresponding observations as closely as possible.
We consider \(\boldsymbol \theta\) to be a vector of random variables, which is associated with a prior probability distribution, \(p_{prior}(\boldsymbol \theta)\). Conditioning the prior on observations leads to the posterior probability distribution of the parameters, \(p_{post}(\boldsymbol \theta | \mathbf{d})\), via Bayes’ theorem:
Here, \(\mathcal{L}\left(\boldsymbol \theta | \mathbf{d}\right)\) is the likelihood.
While the Bayesian approach is rather intuitive, we can usually not obtain the posterior analytically - we have to generate samples from the posterior by computing the likelihood for many different samples of \(\boldsymbol \theta\) and each sample requires a model evaluation. Now if each model run is computationally costly, this approach quickly becomes intractable. Multilevel methods can help to alleviate the computational burden of the problem to allow for sampling-based approaches to Bayesian inference with costly models.
Multilevel methods#
The central idea of multilevel methods is simple: instead of computing a Monte Carlo estimate of a quantity of interest (QoI) using a model with high resolution, high accuracy and high computational cost we rely on models with lower resolution, lower accuracy, and lower computational cost to do most of the work.