3.5.1 A New Definition of Causality
As briefly reviewed in Section 2.3.4, recent works have formalised causal hypotheses to make inferences about the extent of a cause. Here we define a new causal assumption tailored to the needs of an IDSS.
Definition 3.5.1. Call an IDSS
D,d0 -determined if Ψθi, i ∈ [m], is a stochastic function of Ψθi(Li,d0), known toGi, for some prescribed decision d0∈D.
Clearly, if this property is part of the CK-class, then this greatly simplifies the learning each panel needs to undertake. Once they have specified their beliefs about the pa- rameters θi under the decision d0, then the panel beliefs under other decisions can be
calculated automatically. Obviously, exactly whether and how this condition is satisfied depends heavily on the domain of the IDSS. Importantly this condition is implicit for classic Perlean CBNs in the following sense.
Lemma 3.5.2. A CBN with vertex set{Yi :i∈[n]} in a given CK-class is a
D,d0 - determined IDSS whenever d0 is the decision not to intervene but to simply observe the system, D consists of Perlean interventions, and panel Gi delivers beliefs about θBi,
i∈[m], where B1, . . . , Bm is a partition of [m].
Proof. This result is a direct consequence of the definition of a CBN given in Defini- tion 2.3.50. Since the decision space consists of Perlean interventions only, each belief Ψθi(Li,d) = Ψθi(Li,do(YA=yA)), A⊂[n], either coincides with the probabilities in the
idle system, coinciding with A = ∅, or consists of the degenerate distributions having mass one at the manipulated values. Therefore this is a{D,d0}-determined IDSS.
It can be checked that, for instance, causal hypotheses for CEGs or MDMs provide a
D,d0 -determined IDSS, where
D,d0 are analogously defined. Within the context of IDSSs it is helpful to extend the usual causal assumptions since the collective might want the flexibility not to map the effects of enacting a Perlean do operation only, but also more complex decision rules, thus embellishing D. Furthermore, the natural comparatord0 for predicting what might happen, might not simply consist of observing the idle system, but rather following routine procedures or past protocols. Therefore our condition provides the basis for more comprehensive analyses which are not simply based on the assumption of the existence of a non-manipulated system.
3.5.2 Admission of Experimental Evidence
Different panels in most IDSSs will want to accommodate not only observational data, in ways discussed in Section 3.4, but also experimental one. Here covariates are con- trolled and set to certain values and the effect on a response variable is observed. So, for instance, in a nuclear IDSS one module might predict the radiation absorbed by different types of food stuffs when exposed in a controlled and measured way in a lab- oratory to certain types and durations of radiation. Implicitly, the causal hypothesis is then adopted which assumes that these absorption distributions would be the same were these plants exposed to the same types, durations and levels of exposure on the ground in observational settings. In an actual accident the panel in charge of this absorption module would receive the stochastic process of radiation data from a dispersion/depo- sition module and match this to laboratory settings at the same values. Essentially, this demands that the parameters in the observational setting can be identified with the parameters in the controlled one.
So most operational IDSSs need to assume causal hypotheses relating controlled experi- ments and the unfolding disaster. Recall that Daneshkhah and Smith [2004] showed that in the context of BNs the panel independence assumption necessary to ensure an IDSS’s distributivity is intimately linked and plausible only when certain causal hypotheses can be entertained.
Here, just as in the observational case, we introduce a class of likelihoods associated to data collected in designed experiments that can allow for distributed Bayesian updating.
Definition 3.5.3. Call an IDSS e−panel compatiblefor a collection of datasetsxe=
(xe1, . . . ,xep)T obtained from experiments e = (e
i)Ti∈[p] if the likelihood associated to e can be written as
le(θ|xe) = Y
i∈[m]
where lei(θi |tei(xe))is a function of the parameter vector θi overseen by panel Gi only,
i∈[m], and tei(xe) is a statistic of xe.
By far the most common suite of such experiments is one composed of collections of inde- pendent experiments that can be partitioned across the panels, where each experiment is informative only about the parameters overseen by a particular panel. However a single experiment may be informative for parameters overseen by different panels. If such an experiment were orthogonal over the vectors of parameters under the responsibility of different panels, then the orthogonality would ensure that the likelihood separates across these vector parameters. Therefore, e-panel compatibility would still hold in this case.
Again an important special case that automatically implies this separation is when the overarching qualitative structure is a CBN under experimental manipulations.
Lemma 3.5.4. Suppose an IDSS is
D,d0 -determined, where d0 ∈ D consists of not intervening in the system. Then the IDSS is e-panel compatible for a collection of experiments e= (ei)Ti∈[p], if e consists of independent randomised designed experiments
ek, k∈[p], that observe the response of a variable Yk in the vertex set of a CBN, where
each design consists of the Perlean intervention do(YΠk =yΠk), for yΠk ∈YΠk.
Proof. This is a straightforward consequence of the definition of a CBN in Definition 2.3.50. In this setting this implies that theθk’s appearing in the manipulated experiment
are equal to Gk’s corresponding parameters of interest in the observational setting,
coinciding tod0 ∈D. But because this is so for the idle controld0, it is also true for all
d∈D since the IDSS is
D,d0 -determined.
Now that we have introduced classes of experiments associated to likelihoods exhibiting the type of separation required for distributivity, we show that Bayesian updating can be distributed to panels in the case the structural consensus includes a CBN.
Theorem 3.5.5. Suppose the CK-class of a
D,d0 -determined IDSS includes a CBN, whered0 consists of not intervening in the system. Suppose panel independence is in the CK-class and panels oversee separate components of the parameter vector θ. Suppose data admitted in the IDSS at any time t is of the form of both a collection of ancestral random samples of that CBN and a collection of randomised designed experiments as in Lemma 3.5.4. Suppose all these data sets are independent of each other. Then if the IDSS was distributed and sound a priori it remains so at time t.
Proof. Since all different datasets are independent of each other, the likelihood of the arriving data is simply the product of the likelihoods associated with each component
sample/experiment. So it is sufficient to show that each component likelihood is either panel separable or e-panel compatible. Now Lemma 3.5.4 guarantees that the data arising from these experiment is e-panel compatible and therefore exhibit the required separability. At the same time Proposition 2.3.13 guarantees that the likelihood associ- ated to the random sample is panel separable.
Again analogues of this theorem are also true for a CK-class of an IDSS which contains different overarching qualitative models such as the CEG or the MDM. In particular, Bayesian updating can be distributed to panels whilst retaining the soundness of the IDSS whenever the causal assumptions underpinning the IDSS are such that parameters in the observational setting can be identified with parameters in the controlled one, and the associated likelihoods from these experiments are e-panel compatible.