How to Prove Inconsistency Even if Some Parameters Are Highly Uncertain

How to Prove Inconsistency Even if Some Parameters Are Highly Uncertain 

The presumption of the approach above is that all the model parameters that significantly affect the model predictions Mt are known with reasonable accuracy, so it is not completely unreasonable to assume that the errors in the M, are normally distributed (although a normal distribution is only reasonable if the model prediction could vary over the full range -oo to +oo for bounded quantities appropriately bounded distributions should be used). However, very often some of the important model parameters are not well established from prior work, so we do not know if their uncertainties are normally distributed. In fact, often it would be most reasonable to use the experimental data to try to determine these highly uncertain parameters x. 

When experimental data is unavailable the critical question becomes is there any physically reasonable choice of these very uncertain parameters x that makes the model and the data consistent If not, we can be confident that the model and the data are inconsistent, and if we trust the data and the estimated error bars we can reasonably conclude that the data have disproved the model. Mathematically, we are confident the model and the data are inconsistent if and only if 

Where Y(t x) is not known explicitly, but only implicitly as the solution of a system of stiff differential or DAEs such as Eq. (7) which depend parametrically on x, and x is confined to lie in its physically reasonable range. The functional M( that relates the recorded experimental measurement M to the underlying species concentrations and other state variables could in principle be quite complicated, but often it is linearly dependent on just a few of the species concentrations at some time measurement- As discussed in a later section, our estimate of the error bar in the model prediction for a fixed x might depend on x, that is why ut(x) is written in Eq. (21). 

While this is conceptually simple, numerically it is very difficult to find the global maximum of P, because of the fact that the dependence of Y on x is only known implicitly through a complicated numerical procedure (solving the whole simulation). 

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