Philip A. Leicester

An explication of uncertain evidence in Bayesian networks: likelihood evidence and probabilistic evidence

This paper proposes a systematized presentation and a terminology for observations in a Bayesian network. It focuses on the three main concepts of uncertain evidence, namely likelihood evidence and fixed and not-fixed probabilistic evidence, using a review of previous literature. A probabilistic finding on a variable is specified by a local probability distribution and replaces any former belief in that variable. It is said to be fixed or not fixed regarding whether it has to be kept unchanged or not after the arrival of observation on other variables. Fixed probabilistic evidence is defined by Valtorta et al. (J Approx Reason 29(1):71–106 2002) under the name soft evidence, whereas the concept of not-fixed probabilistic evidence has been discussed by Chan and Darwiche (Artif Intell 163(1):67–90 2005). Both concepts have to be clearly distinguished from likelihood evidence defined by Pearl (1988), also called virtual evidence, for which evidence is specified as a likelihood ratio, that often represents the unreliability of the evidence. Since these three concepts of uncertain evidence are not widely understood, and the terms used to describe these concepts are not well established, most Bayesian networks engines do not offer well defined propagation functions to handle them. Firstly, we present a review of uncertain evidence and the proposed terminology, definitions and concepts related to the use of uncertain evidence in Bayesian networks. Then we describe updating algorithms for the propagation of uncertain evidence. Finally, we propose several results where the use of fixed or not-fixed probabilistic evidence is required.

From Information to Evidence in a Bayesian Network

Evidence in a Bayesian network comes from information based on the observation of one or more variables. A review of the terminology leads to the assessment that two main types of non-deterministic evidence have been defined, namely likelihood evidence and probabilistic evidence but the distinction between fixed probabilistic evidence and not fixed probabilistic evidence is not clear, and neither terminology nor concepts have been clearly defined. In particular, the term soft evidence is confusing. The article presents definitions and concepts related to the use of non-deterministic evidence in Bayesian networks, in terms of specification and propagation. Several examples help to understand how an initial piece of information can be specified as a finding in a Bayesian network.

Probabilistic evaluation of UK domestic solar photovoltaic systems: An integrated geographical information system PV estimation tool

It is shown how key predictor parameters for the spatial estimation of PV yield, self -consumption and thereby economic and social indicators can be extracted from a GIS system and introduced into a Bayesian Network model. This model endogenises the uncertainties and incorporates spatial variability inherent in these parameters. Empirical monthly and annual yield measurements obtained from over 600 PV installations have been obtained and compared with estimated yields obtained by two key solar tools used for performance estimation in the UK – these are PVGIS and the UK Government’s Standard Assessment Procedure (SAP) for domestic buildings. Mean bias estimates and root mean square error estimations were obtained for each tool and the results used to construct an uncertainty distribution in PV yield prediction given key input parameters such as system rating, orientation and tilt. This uncertainty was used to furnish a probabilistic graphical model with a prior distribution for PV yield estimation. This was integrated into a Geographical Information (GIS) system furnished with roof and building stock parameters including roof attributes obtained from lidar data. Elements held in a vector layer of the GIS system can be selected and the resultant distributions of input parameters automatically fed to the model to yield a posterior distribution of the PV yield. The model is able to propagate the yield uncertainty to other probabilistic models, including ones which predict the internal rate of return and self -consumption. The latter is in turn predicted by empirical marginal distributions of domestic electricity consumption. Thus with a given posterior distributions of PV yield, new posterior distributions for the internal rate of return, self-consumption and carbon emission savings are automatically calculated. By integration with GIS this novel approach allows the spatial analysis of the uncertainty pertaining to representative risk factors for PV adoption in the UK, and facilitate the estimation by installers, investors, and local authorities in a manner which endogenises uncertainty.