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for using the squared error (see, for example, Halpern, 2003), but once it is realised that it is a
choice there is no real reason why it should be an exclusive choice. This wider view does open
up the possibility of using
likelihood measures
based on different types of model evaluation,
including qualitative and fuzzy measures that might reflect the complex sources of error in the
modelling process in different ways.
The definition of a likelihood measure for use within the GLUE methodology is more relaxed.
It requires only a measure that is zero for nonbehavioural models and increases monotonically
as performance or goodness of fit to the available observational data increases. There are
many different measures that could be used (including performance measures that have been
used in optimisation of model parameters in the past). The choice of measure should reflect
the purposes of the study and the nature of the errors being considered but, ultimately, the
choice of a likelihood measure is subjective. In this, it has much in common with recent
developments in approximate Bayesian computation (ABC) that allow for model conditioning
without likelihoods using rejection criteria (see, for example, Beaumont
et al.
, 2002, 2009;
Toni
et al.
, 2009).
B7.1.2 Hierarchical Statistical Likelihoods
Recent developments in model conditioning using statistical theory have been concerned with
trying to take more explicit account of different sources of uncertainty, rather than dealing only
with the lumped model residuals of Equation (B7.1.2) (e.g. Kuczera
et al.
, 2006; Liu and Gupta,
2007; Gotzinger and Bardossy, 2008; Huard and Mailhot, 2008; Vrugt
et al.
, 2008; Thyer
et al.
,
2009; Renard
et al.
, 2010). This requires a hierarchical Bayesian approach to identification with
the specification of separate error models for input uncertainties, observation uncertainties,
parameter uncertainites and residual uncertainties. These specifications will have parameters
(often called
hyperparameters
of the analysis) that are identified as part of the conditioning
process. A typical assumption for input error, as shown in Section 7.8, is that the uncertainty
associatedwith the rainfall estimation for an event can be treated as amultiplier on the observed
rainfalls, where the multipliers are chosen from a log normal distribution defined by mean and
variance hyperparameters (e.g. Thyer
et al.
, 2009; Renard
et al.
, 2009; Vrugt
et al.
, 2008).
Errors in discharge observations might be represented by a heteroscedastic model in which an
additive error is assumed to have a variance that is proprtional to the value of discharge. The
simplest heteroscedastic model has a single hyperparameter, a constant relative standard error
(Huard and Mailhot, 2008; Thyer
et al.
, 2009; McMillan
et al.
, 2010) but more complex models
might also be defined (Moyeed and Clarke, 2005). It might also be necessary to allow that the
rating curve is non-stationary in mobile bed rivers that are subject to extreme discharges (e.g.
Westerberg
et al.
, 2010a). A residual variance is usually assumed to follow the simple additive
form of Equation (B7.1.2).
Note that a number of additional degrees of freedom have been added here in explaining
the uncertainty in the total model residuals, while constraining the representation of different
sources of uncertainty to specific assumed forms. Such a disaggregation will, ideally, assign the
effects of different sources of uncertainty correctly but, as with any form of statistical inference
of this type, the validity of the inference will depend on the validity of the assumptions. In
this case, it might be quite difficult to test assumptions about the sources of uncertainty and
their stationarity independently of the analysis and, as noted in Section 7.8, it is possible that
the sources of uncertainty interact; rainfall multipliers may interact with model structural error,
for example. Where the assumptions are not strictly valid, this will almost certainly lead to
over-conditioning of the posterior parameter distributions.
B7.1.3 Informal Likelihood Measures
It follows from this discussion that treating all sources of error as aleatory tends to result in
overconditioning of posterior model likelihoods when some epistemic sources of error are
