Geoscience Reference
In-Depth Information
A Bayesian analysis requires three basic elements:
a definition of the prior distribution for the uncertain quantities to be considered (in some cases,
“noninformative” priors might be defined where there is no prior knowledge about a quantity);
a definition of the likelihood function that reflects how well a model can predict the available obser-
vations;
a method for integrating the product of prior probabilities and likelihoods to calculate the posterior
distribution.
The definition of the prior is much the same requirement as for a forward uncertainty estimation (see
Section 7.5). The definition of the likelihood depends on making formal assumptions about the structure
of the residuals (see Box 7.1). The third requirement is generally satisfied now by the use of different forms
of Monte Carlo Markov Chain (MC 2 ) methods (see Box 7.3). It is equvalent to trying to search for the
high likelihood areas of the posterior parameter space without knowing where they might be beforehand
(though the definition of the prior is intended to be helpful in this respect). To do so efficiently means
concentrating the effort on those areas where past samples have indicated high likelihoods (but with
some additional random sampling to try to avoid missing areas of high likelihood that have not yet been
sampled, especially when a large number of parameters might be involved). MC 2 methods provide tools
for making this type of sampling efficient. For complex problems, this still requires tens or hundreds of
thousands of runs of a model. As with the hill-climbing optimisation techniques, they work best when
the dimensionality of the space is low and the surface is simple in shape.
The critical issue with the application of Bayesian methods to rainfall-runoff models is really the
definition of the likelihood function. This has been the subject of significant debate in the hydrologi-
cal literature (Beven and Young, 2003; Mantovan and Todini, 2006; Beven, 2006a, 2008; Hall et al. ,
2007; Montanari, 2007; Todini and Mantovan, 2007; Beven et al. , 2008; Stedinger et al. , 2008; Beven
and Westerberg, 2011). It involves questions of belief about the nature of errors (and therefore dif-
ferences in philosophy). A formal definition of a likelihood function requires the specification of a
statistical model of different sources of an error (though often applied only to the residuals between ob-
served and predicted values). Different assumptions about the errors lead to different likelihood functions
(see Box 7.1). Generally this also requires assuming that the statistical model has the same structure for
all the rainfall-runoff models tried, and that the structure does not change through time. The objectivity
of the approach is then predicated on two conditions. The first is that if these assumptions are valid, then
the probability of predicting an observation conditional on the model is mathematically justified. The
second is that the validity of the assumptions can be checked against the characteristics of the actual
model residuals (and the error model changed, if necessary).
This check for the validity of the assumptions to ensure that an appropriate error model is being used
is, therefore, an important part of the process. In practice, it should really be an iterative process since,
under the assumption that there is a best possible model (even if not a “true” model because all models
are approximations), it is the structure of the errors of that maximum likelihood model that must be
checked, but finding the maximum likelihood model depends on defining a likelihood function for an
error structure. There are many examples in the hydrological literature of modelling studies using the
Bayesian framework where it is only too obvious that the likelihood function that has been used is not
valid (e.g. Thiemann et al. , 2001; Feyen et al. , 2007). This is bad practice, in that the resulting posterior
parameter distributions will be biased. In fact, even small departures from the validity of the assumptions
can lead to biased parameter inference (Beven et al. , 2008a).
Experience suggests that hydrological models do not, in general, conform well to the requirements of
the classical techniques of statistical inference. This is, at least in part, because so many of the uncertainties
encountered are the result of lack of knowledge rather than the result of random realisation effects (i.e.
epistemic rather than aleatory errors, as noted earlier in this chapter). If epistemic errors are treated as if
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