A first example

Usually, the sole use of a simple parametric model consisting of precise probabilities is hardly justifiable in real applications because real data almost never stem from such a model and, if they would do so, we could not be sure that they really do. It is a well known fact that small deviations from a precise model can have large effects on the statistical methods – cf. e.g. Huber (1981). However, a precise model is usually not precisely true. This is one reason for the use of imprecise probabilities. Of course, it is far more easy to determine upper and lower bounds for the probabilities than to determine precise probabilities. Tough it will not be possible to precisely determine correct upper and lower bounds, small changes in the upper and lower bounds should only have small effects in the statistical evaluation. This is usually true but, unfortunately, this is not always true. Arbitrarily small changes in the upper and lower bounds can have arbitrarily large effects in some cases. This is because the theory of imprecise probabilities commonly uses a method which is potentially most instable – namely natural extension.

Especially in applications, the method of natural extension is a frequently used comfort- able tool because it enables to define a coherent upper prevision P onL∞(X,A) or on a subset ofL∞(X,A) in the following way: An experimenter determines an upper prevision

P : K → _R

on some subsetK ⊂ L∞(X,A) ofL∞(X,A) and then extends this prevision to a coherent lower prevision on a larger set – by means of a natural extension, cf. Section 2.3. For simplicity of notation, we may assume that he extends the prevision to the whole set L∞(X,A).

In general, such a proceeding can be very instable and may lead to arbitrary results. To see this, let us consider the following simple example:

Put X = [0,1] and let A be the Borel-σ-algebra of [0,1]. Put f0 : [0,1] → _R, x 7→ x and K={f0}. Furthermore,

P[f0] = 0 and

P0[f0] =ε

where 0< ε <1. Then, the natural extensions are given by P[f] = sup P∈M P[f] ∀f ∈ L∞(X,A) and P0[f] = sup P0_∈M0 P0[f] ∀f ∈ L∞(X,A) where M = P ∈ba+₁(X,A) P[f₀] = 0 M0 = P0 ∈ba+₁(X,A) P0[f₀]∈[0, ε]

For every P ∈ M, it follows from f0 ≥ εI[ε,1] that 0 = P[f0] ≥ P εI[ε,1] = ε PI[ε,1] ≥ 0 (5.1) and therefore, PI[ε,1] = 0 . Hence, PI[ε,1] = 0

Let δε be the Dirac measure in ε. Thenδε[f0] =ε impliesδε ∈ M0 and

1 = sup x∈[0,1] I[ε,1](x) ≥ sup P0_∈M0 P0 I[ε,1] ≥ δε I[ε,1] = 1 Hence, P0I[ε,1] = 1 Summing up, we have

PI[ε,1] = inf x∈[0,1]I[ε,1](x), P 0 I[ε,1] = sup x∈[0,1] I[ε,1](x)

This is, indeed, the worst thing that can happen. The unpleasant message of this example is:

Determining a coherent upper prevision on some functions K ⊂ L∞(X,A) in

a first step and extending the coherent upper prevision (by natural extension) to some functionsf ∈ L∞(X,A)in a second step may lead to arbitrary results: Arbitrarily small changes of one upper bound

P[f0]

(where f0 ∈ K) may have arbitrarily large effects on the bounds P[f], f ∈ L∞(X,A)

Note that the above example is not a pathological one: The sample space is a compact interval in _R, the algebra A is the Borel-σ-algebra and the coherent upper prevision on K is a very easy one because K only consists of one element f0 and this f0 is a linear function. It would even have made no difference if we would have takenKto be the linear space

K := {af0| a∈R}

However, the above example, indeed, is somehow special because we have P[f0] =P[f0]

and this is a precise prevision which is not really what we want in imprecise probabilities. Nevertheless, the use of such imprecise probabilities where

P[f] = P[f] (5.2)

at least for some (non-constant) functionsf ∈ Kis not unusual in applications of imprecise probabilities. Though such precise values (5.2) of imprecise probabilities seem to be problematic, this problem does not arise in classical probability theory (where upper and

lower bounds always coincide) because, there, it is not possible to change single values in the above manner and something like a method of natural extension does not exist anyway.

One might argue that the coherent upper prevision used in the above example is not a good one and that it is enough to take a short look at it in order to see this. However, this example is also an extremely simple one and it is hard to guarantee that such “detecting bad models by a short look at it” still works for more complicated previsions.

The above example does not show that using natural extensions was indefensible in real applications but it shows that natural extension should not be used unthoughtfully. For applications, it would be desirable to have some guidelines which prevent practitioners from arbitrary results because of an instable natural extension. The following subsection makes a first attempt in this direction but it certainly does not succeed in giving a final, satisfactory answer. Hopefully, future research will provide some more insight into this topic.