Uncertainty Quantification of Complex System Models: Bayesian Analysis

Abstract

This chapter summarizes the main elements of Bayesian probability theory to help reconcile dynamic environmental system models with observations, including prediction in space (interpolation), prediction in time (forecasting), assimilation of data, and inference of the model parameters. Special attention is given to the treatment of parameter uncertainty (first-order approximations and Bayesian intervals), the prior distribution, the formulation of the likelihood function (using first-principles), the marginal likelihood, and sampling techniques used to estimate the posterior target distribution. This includes rejection sampling, importance sampling, and recent developments in Markov chain Monte Carlo simulation to sample efficiently complex and/or high-dimensional target distributions, including limits of acceptability. We illustrate the application of Bayes’ theorem and inference using three illustrative examples involving the flow and storage of water in the surface and subsurface. At least some level of calibration of these models is required to match their output with observations of system behavior and response. Algorithmic recipes of the different methods are provided to simplify implementation and use of Bayesian analysis.

Appendix

1.1 A: Derivation of Bayes’ Theorem

Bayes’ theorem (also referred to as Bayes’ law or Bayes’ rule) is a relatively simple but fundamental result of probability theory that allows for the calculation of certain conditional probabilities. The theorem specifies the relationship between the probability of two entities, A and D, or P(A) and P(D), and their respective conditional probabilities, P(A|D) and P(D|A). This theorem follows logically from Kolmogorovs (1903–1987) axiomatic definition of probability and is consistent with the frequentist and subjectivist approach to epistemology.

If P(A) > 0 and P(D) > 0 denote the probability of two different events A and D, then the conditional probability of A given event D is equivalent to

$$ P\left(A|D\right)=\frac{P\left(A\cap D\right)}{P(D)}, $$

(83)

where P(A ∩ D) signifies the probability of the union of events A and D. Similarly, the conditional probability of D given event A is

$$ P\left(D|A\right)=\frac{P\left(D\cap A\right)}{P(A)}. $$

(84)

Per definition, P(A ∩ D) = P(D ∩ A) which implies that

$$ P\left(A\cap D\right)=P(D)P\left(A|D\right)=P(A)P\left(D|A\right), $$

(85)

which after simple rearrangement leads to Bayes’ theorem in Eq. (11)

$$ P\left(A|D\right)=\frac{P(A)P\left(D|A\right)}{P(D)}, $$

(86)

where A signifies the parameter values, and B denotes the data.

The different probabilities in Eq. (86) may have different interpretations, depending on the intended goal of application. Within the context of statistical inference, Bayes’ theorem expresses mathematically how a subjective initial degree of belief, P(a), in a proposition a, changes rationally to P(a|d) in response to new data, d. The evidence d is not to be confused with the evidence, P(d), or marginal likelihood. The term P(a) is called the prior distribution, and P(a|d) denotes the posterior probability density function, or the degree of belief having accounted for the evidence d. This subjectivist approach is the cornerstone of Bayesian inference, yet Bayes’ theorem has much wider applicability. In our application of Bayes theorem, we can replace the conditional probability, P(d|a) with the data likelihood, L(a|d) as all our inferences are drawn from the residuals of a and d, that is a − d. Whether we center the measurement error distribution on the proposition, a (simulated data), or the “evidence,” b (observed data), this does not change the degree of belief in a.

To illustrate the application of Bayes theorem, let us consider a simple thought experiment in which we are trying to infer the probability that someone has a disease, X, given that they have some symptom, S. That the person has this symptom is clearly visible to the eye, but whether they have the decease is not evident. Bayes’ law tells us that P(X|S) can be derived from

$$ P\left(X|S\right)=\frac{P(X)P\left(S|X\right)}{P(S)}. $$

(86)

So to compute P(X|S), we need to know the prior probability, P(S) and P(X), of the symptom and the disease, respectively (how common are the symptom and disease), and P(S|X), the probability that someone has symptom S given that he/she has the disease, X (via lab tests).

We can confirm Bayes’ law with a simple practical example (see Fig. 23) wherein the occurrence of two, presumably correlated, events, Rain and Thunder, is observed at some place on our planet for a period of n = 20-days. The prior probabilities of both events, P(R) and P(T), and their respective conditional probabilities, P(R|T) and P(T|R), are easy to calculate from the data.

Fig. 23

A simple data set of two discrete events, Rain and Thunder. A value of one (zero) on a given day means that the event has (not) been observed. The data can be used to calculate the prior probabilities of R and T, and their conditional probabilities, P(R|T) and P(T|R). These conditional probabilities can be confirmed with Bayes’ law

We can now use these (conditional) probabilities to benchmark Bayes’ law. Indeed, per Bayes’ theorem, P(R|T) = P(R)P(T|R)/P(T), which gives P(R|T) = (11/20 × 4/11)/(5/20) = (4/20)/(5/20) = 4/5. This confirms the value of P(R|T) = 4/5 computed directly from the data. The same result is found for P(T|R).

1.2 B: MATLAB Code DREAM

The core of the DREAM algorithm can be written in MATLAB in about 30 lines of code (see below). Based on input arguments, prior (function handle that draws samples from prior distribution), f (function handle that returns target density), N (number of chains), M (desired number of iterations), d (number of parameters) and problem (structure that allows user to pass additional input arguments to f or prior), the DREAM function returns as output to the user the multidimensional array x which stores the sampled states and corresponding target densities of each chain (third dimension). Example function handles for some d-variate uniform prior distribution, prior = @(N,d) unifrnd(−10,10,N,d), and standard normal target distribution, f = @(x,d) mvnpdf(x,zeros(1,d),eye(d)).

The variables in the DREAM function are chosen carefully to match, insofar possible, their symbols used in the main text. Built-in functions are highlighted with a low dash and information about their respective input and output arguments is provided by the MATLAB “help” utility. The jump vector, dx(j,1:d), of the jth chain contains the desired information about the scale and orientation of the proposal distribution and is derived from the remaining N-1 chains. The function check() is used as a patch for outlier chains, a critical vulnerability of multi-chain MCMC methods such as SCEM-UA, DE-MC, and DREAM (Vrugt et al. 2003; ter Braak and Vrugt 2008; Vrugt et al. 2008, 2009). Note, vectorization of the inner (proposal) loop would enhance significantly computational efficiency, yet affects negatively readability.

The MATLAB code of DREAM presented in this Appendix has several important restrictions (e.g., uniform prior, fixed crossover selection probabilities, convergence is not monitored, lack of built-in likelihood functions) – all of which are addressed in the MATLAB toolbox of DREAM developed by Vrugt (2016). This toolbox also allows the user to evaluate the chains in parallel using the distributed computing toolbox of MATLAB (see Vrugt (2016) for a multicore DREAM implementation). This parallel implementation violates detailed balance of the sampled chain trajectories, nonetheless, benchmark experiments on a diverse set of problems have shown that this violation hardly affects the results.

1.3 C: MATLAB Code DREAM_(LOA)

Basic MATLAB implementation of the DREAM_(LOA) algorithm for limits of acceptability sampling. This code is identical to DREAM in Appendix B except for lines 29–31 which accommodate the revised acceptance rule of Eq. (80). Input argument f is an anonymous function handle that returns the value of the fitness in Eq. (81), and the structure problem allows the user to pass to f the observed data and corresponding limits of acceptability.

The variables in the DREAM_(LOA) function are chosen carefully to match, insofar possible, their symbols used in the main text. Built-in functions are highlighted with a low dash. Detailed documentation on each of these functions can be found in the MATLAB “help” system.

The fitness has to be defined as an anonymous function handle as follows, f = @(x,problem) fitness(x,problem), wherein problem is a structure with fields Yobs and Delta that store in a n-vector the measurements and limits of acceptability, respectively. A template fitness function, f, is given below.

The fitness function includes a call to own_model_script, which executes the forward model and returns simulated values for a given d-vector of parameter values, x. This function should be written by the user.