Coherent combination of probabilistic outputs for group decision making: an algebraic approach

Current decision support systems address domains that are heterogeneous in nature and becoming progressively larger. Such systems often require the input of expert judgement about a variety of different fields and an intensive computational power to produce the scores necessary to rank the available policies. Recently, integrating decision support systems have been introduced to enable a formal Bayesian multi-agent decision analysis to be distributed and consequently efficient. In such systems, where different panels of experts independently oversee disjoint but correlated vectors of variables, each expert group needs to deliver only certain summaries of the variables under their jurisdiction, derived from a conditional independence structure common to all panels, to properly derive an overall score for the available policies. Here we present an algebraic approach that makes this methodology feasible for a wide range of modelling contexts and that enables us to identify the summaries needed for such a combination of judgements. We are also able to demonstrate that coherence, in a sense we formalize here, is still guaranteed when panels only share a partial specification of their model with other panel members. We illustrate this algebraic approach by applying it to a specific class of Bayesian networks and demonstrate how we can use it to derive closed form formulae for the computations of the joint moments of variables that determine the score of different policies.

1. The CK-class is similar in nature to the SupraBayesian approach in standard Bayesian combination of subjective distributions approach (French 2011). Here we use the terminology CK-class to emphasize that only some specific information needs to be shared and agreed upon by the different panels of experts.

2. For simplicity, we assume the intercept to be equal to zero since utilities are unique up to positive affine transformations.

3. We think of \(\theta _{0i}'\) as a parameter although this consists of the sum of a parameter \(\theta _{01}\) and an error \(\varepsilon _i\). Note, however, that from a Bayesian viewpoint these are both random variables.

4. Notice that these values are then normalized to give utility functions between 0 and 1.

5. In the multilinear case, higher moments are required. Here we assume that these can be computed from the first two moments using the recursions of normal distributions.

We acknowledge that J.Q. Smith was partly supported by EPSRC Grant EP/K039628/1 and The Alan Turing Institute under EPSRC Grant EP/N510129/1, whilst E. Riccomagno was supported by the GNAMPA-INdAM 2017 Project.

Authors and Affiliations

1. School of Human Sciences and Technology, IE University, Madrid, Spain

   Manuele Leonelli

2. Dipartimento di Matematica, Università degli Studi di Genova, Genoa, Italy

   Eva Riccomagno

3. Department of Statistics, University of Warwick, Coventry, UK

   Jim Q. Smith

Corresponding author

Correspondence to Manuele Leonelli.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix: Proofs

1.1 Proof of Corollary 1

Adequacy is guaranteed if the EU function can be written in terms of \(\mu _{ji}(d)\) and \(k_{{\varvec{b}}}\), \(i\in [m]\), \(j\in [s_i]\) and \(d\in {\mathcal {D}}\). Note that

The argument of this expectation is a monomial of multi-degree lower or equal to \({\varvec{a}}^*\). Moment independence then implies that \(\bar{u}(d)=\sum _{{\varvec{b}}\in B}k_{{\varvec{b}}}\prod _{i\in [m]}\prod _{j\in [s_i]^0}\mu _{ji}(d),\) and the result follows.

1.2 Proof of Theorem 1

Fix a policy \(d\in {\mathbb {D}}\) and suppress this dependence. Under the assumptions of the theorem, the utility function can be written as

Note also that we can rewrite (16) as

where

and \({\mathcal {P}}^m_0([m])={\mathcal {P}}_0([m])\cap \{m\}\). Calling \({\varvec{\theta }}\) the overall parameter vector of the IDSS, the conditional EU function can be written applying sequentially the tower rule of expectation as

From Eq. (17), the definition of a polynomial SEM and observing that the power of a polynomial is still a polynomial function in the same arguments, it follows that

where \(p_m\) is a generic polynomial function. Thus \(\hat{u}({\varvec{Y}}_{[m-1]})+{\mathbb {E}}_{Y_m|{\varvec{Y}}_{[m-1],{\varvec{\theta }}}}\left( \hat{u}({\varvec{y}}_{m})\right)\) is also a polynomial function in the same arguments. Following the same reasoning, we have that

where \(p_{m-1}\) is a generic polynomial function. Therefore the same procedure can be applied to all the expectations in (18). So \({\mathbb {E}}(u({\varvec{Y}})\;|\;{\varvec{\theta }})=p_1({\varvec{\theta }}),\) where \(p_1\) is a generic polynomial function. This defines by construction an algebraic conditional EU, where the functions \(\lambda _{ij}\) are monomials. Quasi independence and Lemma 1 then guarantee score separability holds.

1.3 Proof of Proposition 2

We prove Eq. (10) via induction over the indices of the variables. Let \(Y_1\) be a root of \({\mathcal {G}}\). Thus \(Y_1=\theta _{01}'\), where \(\theta _{01}'\) is the monomial associated to the only rooted path ending in \(Y_1\), namely \((Y_1)\). Assume the result is true for \(Y_{n-1}\) and consider \(Y_{n}\). By the inductive hypothesis we have that, if \(i<j\) whenever \(i\in \varPi _j\),

Note that every rooted path ending in \(Y_n\) is either \((Y_n)\) or consists of a rooted path ending in \(Y_i\), \(i\in \varPi _n\), together with the edge \((Y_i,Y_n)\). From this observation the result then follows by rearranging the terms in Eq. (19). Equation (11) can be proven via the same inductive process noting that \({\mathbb {E}}(Y_1\;|\;{\varvec{\theta }},d)=\theta _{01}'\) and \({\mathbb {E}}(Y_n\;|\;{\varvec{\theta }},d)=\theta _{0n}'+\sum _{i\in \varPi _n}\theta _{in}{\mathbb {E}}(Y_i\;|\;{\varvec{\theta }},d)\).

1.4 Proof of Theorem 2

Under the assumptions of the theorem, the conditional EU function can be written as in Eq. (12). From the linearity of the expectation operator we have that

Applying moment independence and letting \(V_i\) and \(E_i\) be the sets of distinct vertices and edges, respectively, for all the elements \(P\in {\mathbb {P}}_i\), we have that

where \(c_{ik}\) is the element of \({\varvec{c}}_i\) associated to \(\theta _{jk}\) and \(Ch_l\) is the index of a children of the vertex l. The thesis then follows since each of these expectations is delivered by an individual panel.

1.5 Proof of Lemma 3

To prove this result we first show that under the assumptions of the lemma the utility function can be written as

and then prove that

The lemma then follows by substituting into Eq. (20) for \({\varvec{y}}^{{\varvec{a}}}\) given in Eq. (21).

Fix a policy \(d\in {\mathbb {D}}\) and suppress this dependence. We prove Eq. (20) via induction over the number of vertices of the BN. If the BN has only one vertex then

This can be seen as an instance of Eq. (20). Assume the result holds for a network with \(n-1\) vertices. A multilinear utility factorisation can be rewritten as

The first term on the rhs of (22) is by inductive hypothesis equal to the sum of all the possible monomial of degree \({\varvec{a}}=(a_1,\ldots ,a_{n-1},0)\) where \(0\le a_i\le n_i\), \(i\in [n]\). The other terms only include monomials such that the exponent of \(y_n\) is not zero. Letting \({\varvec{n}}_{n-1}=(n_i)_{i\in [n-1]}\), \({\varvec{y}}_{[n-1]}=\prod _{i\in [n-1]}y_i\) and \(u'=\sum _{I\in {\mathcal {P}}_0^n([n])}k_I\prod _{i\in I{\setminus } \{n\}}u_i(y_i)u_n(y_n)+k_nu_n(y_n)\), we now have that

Therefore, Eq. (20) follows from equations (22) and (23). To prove Eq. (21) note that the monomial \({\varvec{Y}}^{{\varvec{a}}}\) can be written as

Equation (21) then follows by noting that

1.6 Proof of Theorem 3

Under the conditions of the theorem, the conditional EU function can be written as in (15). The linearity of the expectation operator than implies that

Applying moment independence and letting \(V_{tot}\) and \(E_{tot}\) be the sets of distinct vertices and edges, respectively, for all the elements \(P\in {\mathbb {P}}=\cup _{i\in [m]}{\mathbb {P}}_i\), we then have that for any \({\varvec{l}}\simeq {\varvec{b}}\)

Score separability then follows since each of these expectations is delivered by an individual panel.

Code for the multilinear factorization

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