Instrumento pretest y postest “concepto del equilibrio químico” (unidad 3)

3.2.1.1. Neighbourhood Models

An important class of models that make use of sets of priors are neighborhood models. These are typically considered in the robust Bayesian approach (see, e.g., Berger et al. 1994; Ríos Insua and Ruggeri 2000), where a certain prior distribution P0 is singled out as a potential model for prior information, but, due to lack of condence in this choice, a neigbourhood around P0 is considered, consisting of distributions `near' P0. The rationale for this approach is to ensure robustness of the Bayesian analysis based on a single prior

P0 by checking that small deviations from P0 do not lead to large deviations in posterior inferences. As mentioned in Section 2.1.3.1, imprecise probability oers a formal, not casu- istic framework for such Bayesian sensitivity analysis; however, interpretation of the sets of priors, and the modelling intention is dierent, especially with respect to the inference situations we perceive as important modelling opportunities for generalised Bayesian infer- ence.25 _{We will thus touch only briey on neighbourhood models, picking out two typical}

examples, although many dierent kinds of neighbourhood models are discussed in the literature (see, e.g., the surveys by Berger, Ríos Insua, and Ruggeri (2000) and Ruggeri, Ríos Insua, and Martín (2005)).

A typical example is theε-contamination class (see, e.g., Berger et al. 1994, Ÿ4.3.2), which

can be informally described as follows: In a (virtual) sample distribution, not all data are distributed according to P0; instead, 100·ε% of the data is distributed according to any

distribution from a setQ, and depending on the choice forQ, a variety ofε-contamination 24_{By some authors, the density ratio class is considered a neighbourhood model where instead of one}

central distribution P0 two distributions are considered (e.g., Pericchi and Walley 1991, Ÿ4.3). We

think, however, that the density ratio class is better characterised as a separate model framework.

25_{These are (i) the possibility of modeling prior near-ignorance (see Sections 2.2.3.2 and 3.1.3), and (ii)}

3.2 Alternative Models Using Sets of Priors 69 classes can be dened.26

Another example for a neighbourhood model is the odds-ratio model. It tries to model approximate adherence to a central probability law with distribution P0 by giving the following constraints for pairs of events A and B:

P(A)

P(B) ≤(1−) P0(A)

P0(B)

, A, B ⊆Ω

The set of distributions P compatible with these restrictions forms then an odds-ratio

model with parameter , and can be represented by a lower prevision E. When such

a model is taken as an imprecise prior in Bayesian inference, the set of posteriors can again be expressed as an odds-ratio model (Destercke and Dubois 2013b, Ÿ7.2). Other neighbourhood models, like, e.g., variants of the ε-contamination class mentioned above,

may instead not be closed under Bayesian updating. 3.2.1.2. The Density Ratio Class

The density ratio class, also known as interval of measures (DeRobertis and Hartigan 1981; Berger 1990), provides also an interesting model framework for Bayesian inference using sets of priors. Here, instead of generating the set of prior distributions by varying their parameters in a set (as in Section 3.1), the set of priors M is dened by bounding the

probability density functions p(ϑ) ∈ M via a lower bounding function l(ϑ) and an upper

bounding function u(ϑ).27

A set of (prior) distributions on ϑ is dened by

Ml,u ={p(ϑ) :∃c∈R>0 :l(ϑ)≤cp(ϑ)≤u(ϑ)} , (3.6) wherel(ϑ)andu(ϑ)are bounded non-negative functions (i.e., non-normalised densities) for

which l(ϑ)≤ u(ϑ) ∀ ϑ. l(ϑ) and u(ϑ) are often called lower and upper density functions,

and only need to be known up to a multiplicative constant. Ifl(ϑ)>0∀ϑ, then (3.6) can

also be written as Ml,u = p(·) : p(ϑ) p(ϑ0₎ ≤ u(ϑ) l(ϑ0₎ ∀ ϑ, ϑ 0 ,

hence the name `density ration class'.

The density ratio class denes a certain type of credal sets; thus, as discussed in Sec- tion 2.1.2, it can also be expressed via an associated coherent lower prevision E_l,u. The 26_{For Q taken as `all distributions', the ε-contamination class is also called `linear-vacuous mixture' in}

the imprecise probability literature (e.g., Destercke and Dubois 2013b, Ÿ7.3), constituting an important special case of coherent lower previsions.

27_{A very accessible presentation of density-ratio classes with parametric bounding shapesl(ϑ)andu(ϑ),}

along with a method for elicitation from an expert providing quantiles (or quantile ranges) for a number of probability values, is given in Rinderknecht, Borsuk, and Reichert (2011). We discuss this model in Section 3.2.3.

70 3. Generalised Bayesian Inference with Sets of Conjugate Priors inExponential Families density ratio class has a number of advantageous properties, especially as compared to many neighbourhood models (see, e.g., Rinderknecht, Borsuk, and Reichert 2011, Ÿ2.3); most importantly, it has the convenient property of invariance under Bayesian updating. The set of posteriors derived from Ml,u through the Generalised Bayes' Rule (i.e., by up- dating element by element, see Sections 2.1.2.5 and 2.1.3) can again be expressed as a density ratio class, with l(ϑ)and u(ϑ) updated according to Bayes' Rule (DeRobertis and

Hartigan 1981).

Although this invariance property is advantageous, a consequence of it is that also the ratio u(ϑ)/l(ϑ) is constant under updating, such that posterior imprecision, as measured

by the magnitude of Ml,u is the same as prior imprecision, for any sample size n (see, e.g., Rinderknecht 2011, Ÿ4.2.2). This is in strong contrast to the behaviour of the models discussed in Section 3.1, where M(0) converges to a one-element set for_{n → ∞}.

It is important to note here that even if the bounding functionsl(ϑ)andu(ϑ)are dened

parametrically (or, as in the approaches by Coolen (1993a; 1994) described below, even as conjugates),Ml,udoes not contain only these parametric densities (or conjugate densities). Instead, Ml,u contains a variety of shapes, where (if l(ϑ) and u(ϑ) are not proportional) the tail behaviour can vary between that of l(ϑ)and u(ϑ).28

The density ratio class is thus similar to sets of priors discussed in the model framework from Section 3.1.1 where M(0) _{is taken as all convex mixtures of parametric priors with} parameters in IΠ(0)_{. Then, also M}(0) _{contains a variety of shapes that, however, do not} allow for substantially dierent tail behaviour, but it also has the same property of invari- ance under Bayesian updating, because M(n) _{can be constructed as the set of all convex} mixtures of distributions with parameters in IΠ(n)_.29