A decision support system for addressing food security in the UK
Martine J. Barons, Thais C. O. Fonseca, Andy Davis, Jim Q. Smith
AA decision support system for addressing foodsecurity in the UK
Martine J. Barons, Tha´ıs C. O. Fonseca, Andy Davis and Jim Q. Smith
Abstract
This paper presents an integrating decision support system to model food security in the UK. Inever-larger dynamic systems, such as the food system, it is increasingly difficult for decision makers toeffectively account for all the variables within the system that may influence the outcomes of interestunder enactments of various candidate policies. Each of the influencing variables are likely, themselves,to be dynamic sub-systems with expert domains supported by sophisticated probabilistic models. Recentincreases in food poverty the UK raised the questions about the main drivers to food insecurity, howthis may be changing over time and how evidence can be used in evaluating policy for decision support.In this context, an integrating decision support system is proposed for household food security to allowdecision makers to compare several candidate policies which may affect the outcome of food insecurityat household level.
Keywords : Integrating decision support systems, Bayesian multi-agent models, causality, coherence,decision support, graphical models, likelihood separation.
This paper gives a proof of concept practical application of the recently developed statisticalintegrating decision support system (IDSS) paradigm. An IDSS is developed for policymak-ers concerned with deciding between candidate policies designed to ameliorate household foodinsecurity within the UK context of rising food charity use.
Food security exists when all people, at all times, have physical and economic access to suf-ficient, safe and nutritious food to meet their dietary needs and food preferences for an activeand healthy life (FAO, 1996). Missing meals and changing diet is a common response to foodinsecurity, and the latter may persist over extended periods, leading to adverse health effects,especially in children (Seligman et al., 2010). Food insecurity can result in an increased risk ofdeath or illness from stunting, wasting, weakened responses to infection, diabetes, cardiovasculardiseases, some cancers, food-borne disease and mental ill health, via insufficient quantity, poornutritional quality of food, contaminated foods, or social exclusion Friel and Ford (2015). Risingfood insecurity has been strongly associated not just with malnutrition, but with sustained dete-rioration of mental health, inability to manage chronic disease, and worse child health (Loopstraet al., 2015a; Loopstra, 2014). Food insecurity is associated with hypertension and hyperlipi-demia which are cardiovascular risk factors. It is also associated with poor glycaemic controlin those with diabetes, whose additional medical expenses exacerbate their food insecurity (Leeet al., 2019). Food insecurity has been found to affect school children’s academic performance,weight gain, and social skills (Faught et al., 2017). Whilst obesity is more prevalent among1 a r X i v : . [ s t a t . A P ] A p r Introduction food-insecure women, controlling for BMI did not attenuate the association of food insecurityand chronic disease (Pan et al., 2012). The recent increases in food insecurity the UK is well known through the much-publicisedincrease in the uptake of humanitarian aid, principally through food banks and their correspond-ing increase in number (Loopstra et al., 2015b). As a nation, the UK is wealthy and one ofthe world’s most food secure; in 2017 it was 3rd of 113, just after Ireland and the USA ( TheEconomist Intelligence Unit, 2019) but by December 2019 has declined to 17th place. In 2013, aletter published in the BMJ (Taylor-Robinson et al., 2013) on the rise of food poverty in the UKalerted readers to the fact that the number of malnutrition-related admissions to hospital haddoubled since 2008/9. When food parcel distribution by the Trussell Trust exceeded one millionin 2014/15, this was interpreted by some as evidence that the UK government is not fulfilling itslegal duty under the International Covenant on Economic, Social and Cultural Rights (UnitedNations Office of the High Commissioner, 1966) to take appropriate steps to realise the rightof everyone to be free of hunger. Year ending March 2019 more than 1.6 million parcels weredistributed, and in the six months to September 2019, the number of parcels had risen by 23%on the previous year (Trust, 2019). Persistent and widespread low pay, the proliferation of zero-hours contracts and rising living costs, especially food prices, have been suggested as contributoryfactors for the increase in food insecurity, and the health consequences of inadequate diets havealso been raised by health professionals (Garratt, 2015). Relative to other advanced westerneconomies, Britain had higher general inflation, higher food, fuel and housing price inflation,lower growth in wages in the years immediately following the 2008 global financial crisis. TheUK also has a history of very large numbers of very low paid employees; many of those accessingfood banks are in work (Field et al., 2014). For many years, the exact scale of the problem in theUK was unknown. This was because there was no systematic, national assessment of the numbersof households experiencing food insecurity, but only small-scale studies (Pilgrim et al., 2012),(Tingay et al., 2003). However, from 2016, the Food Standards Agency included the Adult FoodSecurity Module of the USDA Household Food Security Survey (HFSS) (USDA, 2012) in thebi-annual Food and You Survey. The HFSS contains 10 items for households without childrenand 18 items for households with children (age 0 - 17) to assess their experiences over the last12 months. The HFSS classifies households as being food insecure when the respondent reportsthree or more food insecure conditions and as very low food security category if at least onemember experienced reduced food intake or if insufficient resources for food disrupted eatingpatterns. The latest UK survey, Wave 5 (2018) (Fuller et al., 2019), found that 80% of respon-dents lived in households with high food security, 10% in households classified as marginallyfood secure, and 10% reported living in household with low or very low food security. There ismore food insecurity amongst families with children: those who lived with children under theage of 16 were less likely than those with no children to have high levels of food security (70%compared with 84%). Employment and income are key determinants of food security. Nearly aquarter (23%) of unemployed people lived in households with very low food security, comparedto 4% of those in work. In the lowest income group, 59% of households had high food security,increasing with income to 93% in the highest income households. In households in the lowestincome groups, 13% had very low food security (compared with less than 1% of those in thehighest income households).
Introduction Tab. 1:
Poverty measures across three countries. UK absolute poverty rate measures the fractionof population with household income below 60% of median income in 2010–11, updatedby the Consumer Prices Index. USA Census Bureau uses a set of dollar value thresholdsthat vary by family size and composition to determine poverty. Canada uses the MarketBasket Measure, the concept of an individual or family not having enough income toafford the cost of a basket of goods and services.UK USA CanadaOverall 19.0% 11.8% 9.5%Child Poverty 26.5% 16.2% 9.0%Working adults with no children 16.4% – –Adults 18-64 – 10.7% –Pensioners 13.5% 9.7% 3.9%Food security low (very low) 10.0% 11.1% (4.3%) 12.3% (2.5%)
Like the UK, USA and Canada, are wealthy nations with significant household food inse-curity. In contrast to the UK, the USA and Canada have undertaken regular monitoring ofhousehold food security over many years through the HFSS module within regular householdsurveys (Tarasuk et al., 2016). This means that research on determinants and rates of foodinsecurity over time is more advanced and detailed in USA and Canada than in the UK.The USA and Canada are similar to the UK in their profiles of poverty and types of government,which allows us to draw on their research where UK data and evidence is sparse.In 2017-18, and the UK absolute poverty rate was 19.0%, ranging from 26.5% among childrento 13.5% among pensioners (Bourquin et al., 2019). In the USA, the official poverty rate in 2018was 11.8%, for children under age 18 it was 16.2%, for people aged 18 to 64, 10.7% and forpeople aged 65+, 9.7% (Semega et al., 2019). In Canada, the official poverty rate is 9.5% overalland 9.0% for children. 3.9% of seniors were living in poverty in 2017 (StatCan, 2017), althoughthe Market Basket Measure has been criticised for omitting housing and childcare costs. TheCanadian Low Income measure, 50% of median income, adjusted for family size, was 12.9% in2017 on an after-tax basis. In 2018 in the USA, 11.1% of households were food insecure and4.3% had very low food security. In Canada it was 12.3% in 2011, the latest figures available,with 2.5% of households with very low food security. (Loopstra, 2014; Tarasuk et al., 2010)
There is a need to gather what information does exist for the UK in order to ascertain theprincipal drivers of household food security to support policy-makers to design policy to tacklefood security and to evaluate other policies which may impact on food security.In ever-larger dynamic systems, such as the food security, it is increasingly difficult for deci-sion makers to effectively account for all the variables within the system that may influencethe outcomes of interest under enactments of various given policies. In particular, governmentpolicies on welfare, farming, the environment, employment, health, etc. all have an impact onfood security at various levels. Each of the influencing variables are likely, themselves, to be dy-namic sub-systems with domain expertise, often supported by sophisticated probabilistic models.Within the food system, examples of these are medium to long range weather forecasting whichinfluences food supply which might be large numerical models, and economic models such as au-toregressive or moving average which estimate the behaviour of global markets and prices under
Introduction various plausible scenarios. The emerging crisis in the UK is not merely a matter for charity,but of great concern to policymakers, who are legally and morally obligated to act, but may lackrecent experience in dealing with needs of this kind and scale, and so require decision support.This paper proposes an integrating decision support system (IDSS) (Smith et al., 2015; Baronset al., 2018) for household food security in the UK. The IDSS is a computer-based tool whichintegrates uncertainties of different parts of a complex system and addresses the decision problemas a whole. In Barons et al. (2018), we detail the iterative manner of the development of an IDSS with itsdecision-makers and expert panels. Before the elicitation starts it is always necessary to do somepreparatory work. With the help of various domain experts, the analyst will need to trawl anyrelevant literature and check which hypotheses found there might still be current. We repeatedlyreview the qualitative structure of the IDSS in light of the more profound understanding of theprocess acquired through more recent elicitation. This modification and improvement continuesuntil the decision centre is content that the structure is requisite (Phillips, 1984). Since theprocess of model elicitation is an iterative one, it is often wise to begin with some simple utilitymeasures, proceed with an initial structural model elicitation, and then to revisit the initial listof attributes of the utility; detailed exploration of the science, economics or sociology can promptthe decision centre to become fully aware of the suitability of certain types of utility attributemeasures. By focusing the centre and its expert panels on those issues that really impact onfinal outcomes we can vastly reduce the scope of a potentially enormous model; only thosefeatures that might be critical in helping to discriminate between the potential effectiveness ofone candidate policy against another are required. If there is strong disagreement about whetheror not a dependency exists in the system then we assume initially that a dependency does exist,except where the consensus is the its effect is weak. Further iterations of the model buildingprocess usually clarify the understanding, and if not, a sensitivity analysis can usually distinguisha meaningful inclusion form others. The decision centre also need to decide what time step isthe most natural one to use for the purposes of the specific IDSS. This choice depends on thespeed of the process, how relevant data is routinely collected on some of the components, andsome technical acyclicity assumptions that are typically known only to the decision analysts.There may be conflict between the granularity of informing economic models of the process,sample survey regularity, and the needs of the system. The granularity needed is driven by thegranularity of the attributes of the utility. In addition, decision analysts need to match preciselythe outputs of a donating panel with the requirements of a receiving panel. When these do notnaturally align, then some translation, possibly a bespoke model, may be needed between them.When expert panels design their own systems, sometimes the internal structure of one componentcan share variables with the internal structure of another. So, for example, flooding could disruptboth the production of food and its distribution and yet these might be forecast using differentcomponents. In such cases, the coherence of the system will be lost and the most efficient wayto ensure ongoing coherence is to separate out the shared variables and ask the panels concernedto take as inputs, probability distributions from the expert panel in the shared variable, floodrisk. One element of these IDSS systems is the way they can appropriately handle uncertaintiesassociated with various modules. This is vital to reliable decision making. For example if theinputs from one module are very speculative - and so have a high variance - Then policies thatwork well over a wide range of such inputs will - under the sorts of risk averse decisions we havehere - ted to be preferred to ones whose efficacy is very sensitive to such inputs. That is why weneed conditional inputs to communicate such uncertainties.
Integrating decision support systems Integrating Decision Support systems are introduced in Smith et al. (2015) and Smith et al.(2016) and briefly reviewed in section 2.1. The IDSS aids decision makers in the understandingof a problem by providing a clear evaluation and comparison of the possible options available. Itcombines expert judgement with data for each subsystem resulting in a full inferential procedureable to represent complex systems. However, decision support systems often require sophisticatedarchitectures and algorithms to calculate the outputs needed by the decision-makers to informpolicy selection when the system is composed of many multi-faceted stochastic processes. Thereis currently no generic framework or software which is capable of faithfully expressing underlyingprocesses for the scale of problems under consideration here, nor sufficiently focused to makecalculations quickly enough for practical use in a dynamic, changing environment.In this application, the framework knitting together the different component subsystems inthe IDSS is the dynamical Bayesian Network (West and Harrison, 1997). In particular, the modelcan be seen as a multi-regression dynamic model (MDM) (Queen and Smith, 1993). Here thisframework is extended to allow variances to vary stochastically over time. The assumed approachis suitable because regression models are well understood but we need to allow for the fact thatwithin this application regression coefficients can drift in time. The dynamical model also allowsfor separability of the different components of the series. A simulation algorithm is developedwhich enables decision making to be fast and dynamical over time even for a large system withmany dependent variables and time points with nonlinear characteristics. Using the MDM, wecan model shocks to the system within the given framework by introducing change point. Thissort of property is exploited in the brain imaging (Costa et al., 2019). Within each of the expertpanels lies a complex sub-network of variables. We seem to a BN/DBN for all the modules sincethese are a very well developed method used in main analogous applications and have supportingsoftware easily available. In Section 2.1, the integrating decision support system methodology isbriefly reviewed. Section 3 details the model and variables used for utility computation in thecontext of food security in the UK. Then Section 4 presents the outputs and policy evaluationfor the food security system. We end the paper with a short discussion of our findings and theplanned next steps in this research programme.
In this section, we briefly review these recent methodological developments to support in-ference for decision support as they apply here. Full details and proofs are provided in (Smithet al., 2016).Consider a vector of random variables relevant to the system Y = ( Y , . . . , Y n ). Typically,there are expert panels with expertise in particular aspects of the multivariate problem. Themost appropriate expert panels for each sub-system are identified, each sub-panel will defer tothe others, adopting their models, reasoning and evaluations as the most appropriate domainexperts. Each expert panel, G i , is responsible for a subvector Y B i of Y , with B , . . . , B m apartition of 1 , . . . , n . The multivariate problem is then decomposed in sub-models. The jointmodel thus accommodates the diversity of information coming from the different componentmodels and deals robustly with the intrinsic uncertainty in these sub-models.Decisions d ∈ D will be taken by a decision maker (DM) where D represents the set of all policyoptions that it plans to consider. In the context of large problems like this, the decision-maker isoften a centre composed of several individuals. These individuals are henceforth assumed to wantto work together constructively and collaboratively supported by using a probabilistic decisiontool that can provide a benchmark evaluation of d ∈ D the underlying processes that drive the Integrating decision support systems dynamics of the unfolding scenario. However, to use the Bayesian paradigm, we would like toassume that this centre will strive to act an a single rational person would when that person isthe owner of the beliefs expressed in the system and so the need for coherence is satisfied. TheDM receives information from each panel and reaches a conclusion that depends on a rewardfunction R ( Y , d ), Y ∈ R Y , d ∈ D . For this level of coherence, we must be able to configure thepanels and their relationships so that certain assumptions are satisfied. Below we briefly outlinewhat these assumptions need to be. More generic descriptions can be found in (Smith et al.,2016).We introduce some notation: For each i = 1 , . . . , m let the subvector Y B i be delivered by G i depend on a function L i ( Y B i ). Each panel G i provides a model Y B i | L i ( Y B i ) , θ B i , d , and priorinformation about θ B i . Each panel G i will deliver summaries denoted by S yi ( L ( Y ) , d ) which areexpectations of functions of Y conditional on the values of L ( Y ) for each decision d ∈ D . Let U ( R ( Y , d )) be the utility function for decision d ∈ D . Our main goal is to compute the expectedutilities { ¯ U ( d ) : d ∈ D} which represents the expected utilities of a decision maker.To be formally valid, any IDSS must respect a set of common knowledge assumptions sharedby all panels and which comprises the union of the utility, policy and structural consensus,described as follows.1. Structural consensus:
The structural consensus requires that all the experts agree, in atransparent and understandable manner, the qualitative structure of the problem in termsof how different features relate to one another and how the future might unfold withinthe system. Formally, these can be couched in terms of sets of irrelevance statements. Wepropose such a structure in 1. There needs to be an agreed narrative of what might happenwithin each component of the system, based on best evidence. Also for each component,there needs to be a quantitative evaluation of how the critical variables might be affectedby the developing environment when appropriate mitigating policies are applied. Wherethere are agreed sets of irrelevance statements, and the semigraphoid axioms are assumedto hold (Smith, 2010), these can be used to populate the common knowledge frameworkbelonging to a decision centre.2.
Utility consensus: requires all to agree a priori on the class of utility functions supportedby the IDSS and the types of preferential independence across its various attributes it willneed to entertain (such as value independence, mutually utility independent attributes(Keeney and Raiffa, 1993) and more sophisticated versions, see Leonelli and Smith (2015).Sections 3.1 and 3.2 give details of the multiattribute utility, its measurement and rationale.3.
Policy consensus: must be sufficiently rich to contain a set of policies that might beadopted and an appropriate utility structure on which the efficacy of these different policiesmight be scrutinised.4.
Adequate:
An adequate IDSS will be able to unambiguously calculate expected utilityscore for each policy that might be adopted on the basis of the panels’ inputs; if it hasthis property the IDSS is called adequate. Note that it should be immediate from theformulae of a given probabilistic composition to calculate these expectations whether ornot the system is adequate (see Smith et al. (2016) for an illustrative example).5.
Sound:
A sound IDSS is one which is both adequate and allows the decision-maker, byadopting the structural consensus, to admit coherently all the underlying beliefs about adomain overseen by a panel as her own, and so accept the summary statistics donated bythe panels to the IDSS.
Integrating decision support systems Distributive:
For such a system to be formal and functional, each component panelcan reason autonomously about those parts of the system they oversee and the centre canlegitimately adopt their delivered judgements as its own. The semigraphoid axioms providemeans to satisfy this requirement and panel autonomy liberates each panel of domainexperts to produce their quantitative domain knowledge in the way most appropriate fortheir domain and using their own choice of probability models. They can update theirbeliefs through any models they might be using and continually refine their inputs to thesystem without disrupting the agreed overarching structure and its quantitative narrative.7.
Separately informed:
An essential condition for panel autonomy is that panel are sep-arately informed. This requirement can be subdivided within a Bayesian framework intotwo conditions - prior panel independence and separable likelihood - using the usual prop-erties of conditional independence. The first of these is a straightforward generalisation ofthe global independence assumption within Bayesian inference (Cowell et al., 1999). Thesecond, the assumption that the collection of data sets gives a likelihood that separates oversubvectors of panel parameters, is far from automatic and is almost always violated whenthere are unobserved confounders or missing data. In such circumstances, one approach isto devise appropriate approximations.8.
Admissibility protocols:
Another approach is to impose an admissibility protocol on theinformation used to make inferences within the system, analogous to quality of evidencerules within Cochrane Database of Systematic Reviews. When data is derived from well-designed experiments, randomisation and conditioning often leads to a likelihood which is afunction only of its own parameters, so trivially separates. When there is a consensus thata quantitative causal structure is a causal
Bayesian network, dynamic Bayesian network,chain event graph or multiprocess model and the IDSS is sound (delegable, separatelyinformed and adequate), then the IDSS remains sound under a likelihood composed ofancestral sampling experiments and observational sampling (Smith et al., 1997).9.
Transparent:
In such a distributive framework, any query made by another panellist oran external auditor can be referred to the expert panel donating the summaries in questionwhich can provide a detailed explanation of its statistical models, data, expert judgementsand other factors informing how its evaluation have been arrived at and why the judgementsexpressed are appropriate.For a distributive IDSS, the question then becomes precisely which information each of thepanels needs to donate about their areas of expertise for the maximum utility scores to becalculated. Provided that the utility function is in an appropriate polynomial form, each panelneed deliver only a short vector of conditional moments and not entire distributions becausethis type of overarching framework embeds collections of conditional independences allowing theuse of tower rule recurrences (Leonelli and Smith, 2015). This facilitates fast calculations andpropagation algorithms to be embedded within the customised IDSS for timely decision-making.In such a system, individual panels can easily and quickly perform prior to posterior analysesto update the information they donate when relevant new information comes to light and thiscan be propagated to update the expected utility scores; this quality is especially useful withindecision support for an emergency, but in any circumstances represents a huge efficiency gainover having to rebuild and re-parameterise a large model. There are a number of frameworkswhich satisfy the requirements of the IDSS properties, including staged trees, Bayesian Networks,Chain graphs, Multiregression dynamic models and uncoupled dynamic BNs.The paradigm outlined here will be illustrated throughout the remainder of the paper througha proof of concept application to an IDSS for government policy for household food security in
Integrating decision support systems the UK, using a Bayesian network as the overarching framework. Bayesian networks (BNs) and their dynamic analogues are particularly suited to the role ofdecision support as they represent the state of the world as a set of variables and model theprobabilistic dependencies between the variables. They are able to build in the knowledge ofdomain experts, provide a narrative for the system and can be transparently and coherentlyrevised as the domain changes.A Bayesian network is formally defined as a directed acyclic graph (DAG) together witha set of conditional independence statements having the form A is independent of B given Cwritten A ⊥ B | C . They are a simple and convenient way of representing a factorisation of ajoint probability density function of a vector of random variables Y = ( Y , Y , . . . , Y n ). Eachnode has a conditional probability distribution, which in the case of discrete variables will beconditional probability tables (CPTs). In this model, L i ( Y B i ) = Y Π i , with Π i the indices ofparents of Y i . The joint density of Y may be written as f ( y | d ) = (cid:89) i ∈ [ n ] f i ( y B i | y Π Bi , d ) . Assume that U ( R ( Y , d )) = (cid:80) i ∈ [ m ] k i U i ( R i ( Y B i , d )). Thus the expected utility is given by¯ U ( d ) = (cid:80) i ∈ [ n ] k i ¯ U i ( d | y Π i ), with¯ U i ( d | y Π i ) = (cid:90) Θ Bi (cid:90) R yBi U i ( R i ( y B i , d )) f i ( y B i | y Π i , θ B i , d ) π i ( θ B i | d ) dy B i dθ B i . Dynamic Bayesian networks are able to accommodate systems which change over time (Deanand Kanazawa, 1989). DBNs are a series of BNs created for different units of time, each BNcalled a time slice. The time slices are connected through temporal links to form the full model.DBNs can be unfolded in time to accommodate the probabilistic dependencies of the variableswithin and between time steps. It is usually assumed that the configuration of the BN does notchange over time, i.e. the dependencies between variables are static.Consider the general setting such that Y it ⊥ Y tQ i | Y t Π i , Y t − i , i = 1 , . . . , n, (1)with { Y t : t = 1 , . . . , T } a multivariate time series composing a DAG whose vertices areunivariate processes and Π i the index parent set of Y it and Y ti = ( Y i , . . . , Y it ) (cid:48) the historicaldata. Thus, the model assumes that each variable at time t depends on its own past series, thepast series of its parents and the value of its parents at time t. This results in the joint densityfunction f ( y ) = T (cid:89) t =1 n (cid:89) i =1 f i,t ( y it | y t Π i , y t − i ) . (2)The observation and system equations are defined as Y it = F it θ it + (cid:15) it ,θ it = G it θ i,t − + ω it , with (cid:15) it ∼ N [0 , V it ] and ω it ∼ N [0 , W it ]. The errors are assumed to be independent of eachother and through time and F it , G it are assumed to be known. Given the initial information, Integrating decision support systems θ i | I ∼ N [ m i , C i ]. The parameters θ it , i = 1 , . . . , n may be updated independently given theobservations at time t . Conditional forecasts may also be obtained independently. These resultsare proved in Queen and Smith (1993) assuming Gaussian distributions for the error terms. Thepredictive density is given by f ( y t | y t − ) = (cid:90) Θ f ( y t | y t − , θ t ) π ( θ t | y t − ) dθ t = n (cid:89) i =1 (cid:90) Θ i g it ( y it | y t Π i , y t − i , θ it ) π i ( θ it | y t − i , y t − i ) dθ it . Let D t = ( y t , D t − ) be the information available at time t. Inference about θ t is based onForward filtering equations to obtain posterior moments at time t .– Posterior distribution at time t − θ i,t − | D t − ∼ N [ m i,t − , C i,t − ];– Prior distribution at time t : θ it | D t − ∼ N [ a it , R it ] , with a it = G it m i,t − and R it = G it C i,t − G (cid:48) it + W it ;– One step ahead prediction: y it | y Π i ,t , D t − ∼ N [ f it , Q it ] , with f it = F (cid:48) it a it and Q it = F (cid:48) it R it F it + V it ;– Posterior distribution at time t : θ it | D t ∼ N [ m it , C it ] , with m it = a it + A it e it and C it = R it − A it Q it A (cid:48) it and e it = y it − f it , A it = R it F it Q − it .If data is observed from time 1 to T then backward smoothing may be used to obtain theposterior moments of θ it | D T , t = 1 , . . . , T . Thus, θ it | θ i,t +1 , D T ∼ N ( h it , H it ) , with h it = m it + C it G (cid:48) i,t +1 R − i,t +1 ( θ i,t +1 − a i,t +1 ), H it = C it − C it G (cid:48) i,t +1 R − i,t +1 G i,t +1 C it and h iT = m iT e H iT = C iT , the initial values.The variance evolution follows West and Harrison (1997) which define V it = V /φ it and φ i,t − | D t − ∼ G ( n i,t − / , d i,t − / φ it | D t − ∼ Gamma ( δ i n i,t − / , δ i d i,t − / , with δ i ∈ (0 ,
1) being the discount factors. The posterior distribution at time t is obtainedanalytically as φ it | D t ∼ Gamma ( n it / , d it /
2) with n it = δ i n i,t − + 1 and d it = δ i d i,t − + S i,t − e (cid:48) it Q − it e it , with S i,t − = d i,t − /n i,t − . This conjugacy results in closed-form recurrenceupdating equations for this variance model. Suppose that θ T was simulated using the Forward filtering and backwards sampling al-gorithm as described in subsection 2.2. The predictive posterior distribution for a replicatedobservation ˜ y is given by f ( ˜y t | y t ) = (cid:90) Θ f ( ˜y t | y t , θ t ) π ( θ t | y t ) dθ t = n (cid:89) i =1 (cid:90) Θ i g it (˜ y it | ˜ y t Π i , ˜ y t − i , θ it ) π i ( θ it | y t Π i , y ti ) dθ it . IDSS: UK Food security The predictive distribution of a new observation ˜ y it may be obtained by simulating from g it ( · | ˜ y t Π i , ˜ y t − i , θ it ). If U ( ˜y t , d ) are linear functions of ˜y t the expected utilities may be computedanalytically using chain rules of conditional probabilities. If U ( ˜y t , d ) is a nonlinear function of ˜y t then expected values are computed by Monte Carlo integration (Robert and Casella, 2004). Notethat some ordering in computing expectations need to be followed, starting from the variablessuch that L i ( Y it ) = ∅ , their descendants and so on.In addition, the types of overarching descriptions suitable for these applications must be richenough to explore both the effects of shocks to the system and the application of policies. Thesecan be conveniently modelled through chains of causal relationships, where causal means thatthere is an implicit partial order to the objects in the system and we assume that the jointdistributions of variables not downstream of a controlled variable remain unaffected by thatcontrol. The downstream variables are affected in response to a controlled variable in the sameway as if the controlled variable had simply taken that value. This is the assumption underlyingdesigned experiments. In every decision support scenario, it is essential to clarify the goals of the decision-maker(DM). Support for household food security is provided in the UK context through Local gov-ernment, typically city or county councils through their financial inclusion and child povertypolicies. The goal of a city or county council in the UK is to fulfil their statutory obligations tothe satisfaction of central government. Whenever possible, they wish to go beyond mere com-pliance and continually improve the lives of the citizens within their geographic region, with aspecial focus on improving the circumstances of the most disadvantaged.In order to construct an IDSS for food security, the next step is to define the utility functionand develop a suitable mathematical form for it. One requirement of the attributes of a utilityfunction is that they must be measurable; it must be possible to say whether an event hashappened or a threshold has been reached. One candidate measure of household food securitywould be data from food bank charities. However, studies have shown that food bank use is nota good measure of food poverty (Kirkpatrick and Tarasuk, 2009; Coleman-Jensen et al., 2016).In the absence of a direct measure of household food security in the UK, the decision-makerneeds a good proxy in order to construct a suitable Utility function. Council officers identifiedthe variables: education, health and social unrest as suitable attributes of a utility.In constructing a utility function based on these attributes, it appeared appropriate to assumevalue independence (Keeney and Raiffa, 1993). Let Z =measures of education, Z =measuresof health, Z =Measures of social unrest, Z =cost of ameliorating policies to be enacted.. Theforms of the marginal utility functions then needed to be specified. For social unrest, healthand education was assumed exponential, whilst the utility on cost was assumed linear. It wastherefore decided that one family of appropriate utility functions might take the form: U ( z ) = a + bz + (cid:88) i =1 − exp ( − c i z i ) , (3)where z = ( z , z , z , z ) and whose parameters ( a, b, c , c , c ) were then elicited. As follows,observable variables are defined as proxies for the attributes required to compute the utilityfunction in (3). IDSS: UK Food security The utility function depends on the proxy variables of health and education which are definedas follows.
Health:
Suppose the expert panellists define a proxy as a function of number of admissionto hospital with diagnosis of malnutrition (primary or secondary) and number of deaths withmalnutrition listed on the death certificate either as primary or secondary cause. Admissionsdata are available in the Hospital Episode Statistics (HES) from the UK government’s Healthand a Social Care Information Service which routinely links UK Office for National Statistics(ONS) mortality data to HES data. In the UK, the number of deaths caused primarily by mal-nutrition are very low and rates are not significantly different over time. Besides, malnutrition isusually accompanied by other diagnoses such as diseases of digestive system, cancers, dementiaand Alzheimer’s disease. Thus, the increase of deaths with malnutrition as a contributory factormight be due to ageing of the population and not due to food insecurity. Regarding admis-sions with malnutrition even the primary diagnosis numbers have increased over time with 391in 2007-08 and 780 in 2017-18. Thus, in this work we considered the primary and secondaryadmission cases as a proxy for the health variable. Thus, the variable Health is defined as thecount of finished admission episodes with a primary or secondary diagnosis of malnutrition codedICD-10. A ICD-10 code of malnutrition on the episode indicates that the patient was diagnosedwith, and would therefore being treated for malnutrition during the episode of care.
Education:
The proxy for education could be defined as a function of educational attain-ment such as the proportion of pupils achieving expected grades in key stages 1, 2 and 4. Eventhough educational attainment is published annually at local and national levels by the UKgovernment’s Department for Education, the score system has changed in previous years andtemporal comparisons are not adequate (Hill, 2014). Thus, as a proxy for education and its re-lation to food security we considered the proportion of pupils at the end of key stage 4 who wereclassified as disadvantaged. Thus, the variable Education is measured as the percentage of pupilsat Key Stage 4 who were classified by the Department for Education as disadvantaged includingpupils known to be eligible for free school meals (FSM) in any spring, autumn, summer, alter-native provision or pupil referral unit census from year 6 to year 11 or are looked after childrenfor at least one day or are adopted from care. Before 2015 this classification considered thosewho have been eligible for Free School Meals at any point in the last 6 years and Children whoare ‘Looked After’. In 2015 this definition was widened to also include those children who havebeen ‘Adopted From Care’. Pupils classified as disadvantaged have a lower average educationalattainment record than other pupils and there is a direct correlation between level of qualifi-cation and unemployment in later life; Poor educational attainment is strongly correlated withteenage pregnancy, offending behaviour, and alcohol and drug misuse. Comparisons between ed-ucational attainment for disadvantage and other pupils indicate a difference of 4.07 (2010/2011)and 3.66 (2016/2017) in the attainment gap index for Key stage 4 for state funded schools inEngland. The gap index are scores measuring the differences between the disadvantaged andnon-disadvantaged groups in Key level 2 and 4 (Hill, 2014). The index is the mean rank for allthe disadvantaged and non-disadvantaged pupils divided by the number of pupils in each cohort.This decimal mean rank difference is scaled to 10 and ranges from 0 to 10, where a higher valuemeans a higher attainment of non-disadvantaged compared to disadvantaged pupils. The indexaims to be resilient to changes in the grading systems and in the assessments and curricula, andmay be used for temporal comparisons.
IDSS: UK Food security Social Unrest:
Inadequate food security can cause food riots (Lagi et al., 2012). In the UK,a riot is defined by section 1(1) of the Public Order Act 1986 as where 12 or more persons whoare present together use or threaten unlawful violence for a common purpose and the conductof them (taken together) is such as would cause a person of reasonable firmness present at thescene to fear for his personal safety, each of the persons using unlawful violence for the commonpurpose is guilty of riot. Riot data is collected by the police. Whilst the likelihood of a foodriot is small in the UK currently, post-riot repairs both to physical environment and communityrelations can be considerable.
Costs
Costs of candidate intervention policies are routinely calculated and form part of thedecision-making process. Indeed, as a response to falling budgets, decision makers might revisethe criteria for assistance of various kinds, for instance by making the eligible cohort smaller.Interventions which are effective but budget-neutral or cost-saving are obviously preferred, how-ever, when the benefit of intervention may not be seen within the same financial year, this wouldform part of the decision-makers’ discussion after the policies had been scored. This is the ap-proach we take here, by scoring the policies and leaving the costs for final discussions of decisionmakers.
Having found a parsimonious form of utility function, we are able to begin to build thearchitecture of the supporting structural model. The paradigm we used for this is describedin detail in (Smith, 2010). The method involves first eliciting those variables which directlyinfluence the attributes of the utility function, then the variables which affect those variablesand so on until a suitable level of detail has been obtained. This was effected using an iterativeprocess, drawing on the food poverty literature and checking with domain experts, refining andrepeating. In particular, the general framework was confirmed by work produced independentlyin Loopstra (2014). The variables and their dependencies for the UK food system are shown inFigure 1.There are a range models which can be used for the overarching model of an IDSS, as listedin Smith et al. (2016), and for the purpose of the IDSS for food security we selected a dynamicBayesian network (DBN) as summarised in subsection 2.2. The structure was assumed to befixed over time.Figure 1 illustrates the 16-node DBN obtained through literature and confirmed by the ex-perts. The node food security represents the two variables, health and education, considered inthe utility function.
IDSS: UK Food security Food securityHousehold income Food costsAccess to credit Benefits Costs of livingHousing (Incl. Energy)Energy costsOil costs Food productionFood importsEconomic context Frost daysEmployment TaxPart−time work
Fig. 1:
IDSS proposed for UK food security decision support.
IDSS: UK Food security Having identified the factors influencing household food security in the UK the next step isto identify the most relevant experts to provide information on these. The panels constitutedfor such an IDSS will often be chosen to mirror the panels that are already constituted forsimilar purposes, e.g. in the UK, the Office for Budget Responsibility, HM Treasury and TheConfederation of British Industry all produce economic forecasts on the UK Economy. Lookingat where the relevant information is held gives some very natural panels.The 16-node DBN illustrated in figure 1 becomes a 9-panel IDSS (figure 2). Panel G2 reportson cost of food given inputs from pane G5 on food supply, incorporating imports and exports,domestic food production and supply chain disruption. Panel G5, in turn, relies on informationfrom G8 the Met office on weather and climate patterns to calculate its expectations of foodsupply, since both domestic and world production and supply chain disruption are weatherrelated. Household income, G1, impacts directly on the utility. Panel G1 relies on informationprovided by G3 and G4 to make its predictions under different policy scenarios. G4 adiviseson cost of living including energy, housing and other essentials. G3 assesses income takinginto account employment, tax and social security, taking inputs from G7 and G9. G7 advisedon demography, including single parents, immigrants, disability and those with no recourse topublic funds. G9 advises on matters of the economy and informs the oil price panel, G6, and thecost of living panel, G4 as well as G3.
Fig. 2:
The expert panels required for this IDSS. Each node represents an expert panel which,using its models and data, provides summaries of expected values and relevant momentsunder each policy decision being considered.
IDSS: UK Food security Here we assume plausible models for the expert panels and utility, based on publicly availabledata.The attributes being measured to compose the food network were obtained at the Office forNational Statistics which publishes official statistics for the UK. The time series for all nodes aremeasured yearly and the temporal window considered goes from 2008 to 2018. Each variable isdetailed at Appendix A.For the purposes of this proof of concept, social unrest was omitted since there was noavailable data. The health and education indicators are the attributes in the utility function andare directly affected by household income (HIncome, panel G ) and food costs (CFood, panel G ). The variables are modelled in the log scale as both are percentages or rates. log ( Health t ) = δ ,t + δ ,t HIncome t + δ ,t CF ood t + (cid:15) ht ,log ( Education t ) = δ ,t + δ ,t HIncome t + δ ,t CF ood t + (cid:15) et . Panel G advises on household income aiming to reflect the amount of money that householdshave available after accounting for the expendures with living (panel G ), taxes and also theaccess to credit and benefits (panel G ). HIncome t = θ ,t + θ ,t Lending t + θ ,t T ax t + θ ,t Benef its t + θ ,t CLiving t + (cid:15) t . The variable costs of food (Panel G ) depends on costs of energy (panel G ) and on foodsupply, imports and exports and food production (panel G ). CF ood t = θ ,t + θ ,t , F P roduction t + θ ,t F Imports t + θ ,t CEnergy t + (cid:15) t . Panel G reports on variables affecting the income such as lending, tax and unemployment.Unemployment depends on the economic context (panel G ) represented by GDP and on part-time workers (panel G ). Lending t = θ ,t + θ ,t U nemployment t + (cid:15) t ,T ax t = θ ∗ ,t + θ ∗ ,t U nemployment t + (cid:15) ∗ t ,Benef its t = θ ∗∗ ,t + θ ∗∗ ,t U nemployment t + (cid:15) ∗∗ t ,U nemployment t = θ ∗∗∗ ,t + θ ∗∗∗ ,t Part-time t + θ ∗∗∗ ,t GDP t + (cid:15) ∗∗∗ t . Panel G reports on costs of living which depend on costs of food (panel G ), on costs ofhousing including energy. Costs of housing depends on costs of energy (panel G ). Cliving t = θ ,t + θ ,t CF ood t + θ ,t CHousing t + (cid:15) t ,CHousing t = θ ∗ ,t + θ ∗ ,t CEnergy t + (cid:15) ∗ t . Panel G (Food supply) reports on food production and imports which depend on the economiccontext (panel G ): F P roduction = θ ,t + θ ,t Gdp t + θ ,t Imports t + (cid:15) t ,F Imports t = θ ∗ ,t + θ ∗ ,t GDP t + (cid:15) ∗ t . Panel G reports on oil costs and energy given inputs from panel G about economic context. COil t = θ ,t + θ ,t GDP t + (cid:15) t ,CEnergy t = θ ∗ ,t + θ ∗ ,t COil t + (cid:15) ∗ t . Model outputs and scenario evaluation Panel G (Demography), G (Weather) and G (Economy) reports on demography, weatherand economic context, respectively with model equations given by log ( P artT ime t ) = θ ,t , + (cid:15) t ,F rost t = θ ,t + (cid:15) t ,Gdp t = θ ,t + (cid:15) t . Using these models as the panels’ models, we now examine what happens to the utility underan number of scenarios.
Figure 3 presents the fit and effects of household income and food costs on health and edu-cation obtained by recursively updating of posterior moments based on the forward filtering andbackward algorithm presented in subsection 2.2. Notice the negative effect of household incomeand positive effect of food costs on the rate of malnutrition and percentage of disadvantagedpupils. Figure 4 presents the fit for all the variables in the food security network.After fitting the dynamical model, different policies were compared using the IDSS approachdescribed in section 2. Policy 1 is ‘do nothing’, i.e. all variables kept on the same observed values.Policy 2 accounts for an increase of 25% in the food costs and policy, such as a no-deal Brexit(Barons and Aspinall, 2020). Policy 3 represents a decrease of 25% in the food costs, such asthrough government subsidies. Figure 5 presents the posterior utility function for the 3 policies.Small values for the utility is associated with smaller rates of malnutrition and smaller percentageof disadvantaged pupils. The expected value of utility for policies 1, 2 and 3 are 0.2400, 0.2808and 0.2091, respectively. Policy 4 considers the situation that food costs are reduced by 15% plushousehold income is increased in 15%, through economic or welfare interventions. In this scenariothe expected utility is 0.2232. Policy 5 is an agricultural policy leading to a reduced the outputof food production (related to prices) by 25% resulting in an expected utility of 0.2161. Notethat the last scenario maintains the variables affecting food production as fixed in the observedvalues and modify the variables lower in the hierarchy such as food costs.
Model outputs and scenario evaluation − . − . − . − . Health year l og ( z t ) − − + − Household income effect on health year d − . . . Food costs effect on health year d − . − . − . Education year l og ( z t ) − − + − Household income effect on education year d − . . Food costs effect on education year d Fig. 3:
Attributes composing the utility function, effects of household income and food costs andMDM fit (mean and 95% credible interval), 2008-2018.
Model outputs and scenario evaluation Household income fit year y t Food costs fit year y t Lending fit year y t Tax fit year y t − . − . − . − . − . Benefits fit year l og ( y t ) Unimployment fit year y t Costs of living fit year y t Costs of housing fit year y t Costs of energy fit year y t Food production fit year y t Imports fit year y t Costs of oil fit year y t Disable and part−time workers fit year y t Frost days fit year y t Gdp fit year y t Fig. 4:
Variables composing the food network and dynamical regression model fit (mean and 95%credible interval), 2008-2018.
Discussion and further developments Utility D en s i t y Policy 1Policy 2Policy 3
Utility D en s i t y Policy 1Policy 3Policy 4
Utility D en s i t y Policy 1Policy 3Policy 5
Fig. 5:
Utility function posterior distribution.
We have shown a proof of concept IDSS for policymakers concerned with ameliorating house-hold food security in the UK. We have identified the main drivers of food security, drawing partlyon research from the USA and Canada where food security has been measured for a number ofyears and therefore understanding of determinants of household food security are more advancedthan in the UK. We have identified plausible expert panels based on UK structures and have con-structed models based on publicly available data. We have demonstrated the output of the IDSSunder a number of policies. We have assumed equal weighting between health and educationalattainment as a proxy for food insecurity. To move form a proof of concept to a working IDSS,we would need to elicit the user preferences for display of the results, as discussed in (Baronset al., 2018).
Description of variables used in the network A Description of variables used in the network
Panel G1 (household income) is represented by the variable HIncome. This variable dependson the household income after expenses.- HIncome: Real net households adjusted disposable income per capita less the final con-sumption expenditure per head.Panel G2 (food costs) is represented by the variable CFood.- CFood: CPI index of 9 food groups, 2015=100. Food costs was measured by a combinationof CPI indices of items representing household dietary diversity (Kennedy et al., 2012).The score is formed by 9 food groups: cereals, meat, fish, eggs, milk, oils and fat, fruits,vegetables and beverages.Panel G3 (income) accounts for access to credit (Lending), tax on the income (Tax), unemploy-ment rate and social benefits.- Lending: Net lending (+)/net borrowing (-) by sector as a percentage of GDP - Householdand non-profit institution serving households.- Tax: Tax on the income or profits of corporations.- Unemployment: Male unemployment rate, aged 16 and over, seasonally adjusted.- Benefits: Social assistance benefits in cash as a percentage of GDP.Panel G4 (costs of living) accounts for expenditure per head (Living) and housing costs (Chous-ing).- CLiving: Consumer price indices of main variables composing the expenditures of a house-hold: Housing, including energy (CHousing), food (CFood), recreation (CRecreation), andtransport (CTransport).- CHousing: CPI of housing, water and fuels.Panel G5 (food supply) accounts for output of food production (FProduction) and imports fromEuropean Union and other countries.- FProduction: Output of food products.- FImports: Food imports from European Union countries plus imports from other countries.Panel G6 (Oil costs) is represented by CPI of fuels and energy (COil and CEnergy):- COil: Liquid fuels, vehicle fuels and lubricants (G) 2015=100.- CEnergy: CPI of energy, 2015=100.Panel G7 (Demography) is represented by part-time work rates (PartTime).- Part-time: Part-time workers ( Ill or disabled).Panel G8 (Weather) is represented by number of days in which the air temperature falls below0 degrees Celsius. In these cases, sensitive crops can be injured, with significant effects onproduction.- Frost: Number of days of air frost.Panel G9 (Economy) accounts for economic context represented by Gross D domestic Product(GDP):- GDP: Gross Domestic Product at market prices, seasonally adjusted.
Description of variables used in the network References
The Economist Intelligence Unit (2019) Global food security index. Web. URL: http://foodsecurityindex.eiu.com/ .Barons, M. J. and Aspinall, W. (2020) Anticipated impacts of Brexit scenarios on UK food pricesand implications for policies on poverty and health: a structured expert judgement approach.
BMJ Open .Barons, M. J., Wright, S. K. and Smith, J. Q. (2018) Eliciting probabilistic judgements for inte-grating decision support systems. In
Elicitation: The science and art of structuring judgement (eds. D. LC, M. A and Q. J), chap. 17. New York: Springer.Bourquin, P., Cribb, J., Waters, T. and Xu, X. (2019) Living standards, poverty and inequalityin the uk: 2019.
Tech. rep. , Institute for Fiscal Studies.Coleman-Jensen, A., Rabbitt, M., Gregory, C. and Singh, A. (2016) Household food security inthe united states in 2015. economic research report no. (err-215) september 2016.
Tech. rep. ,USDA.Costa, L., Smith, J. Q. and Nichols, T. (2019) A Group Analysis using the MultiregressionDynamic Models of fMRI networked time series.
Journal of Statistical Planning and Inference ,43 –61.Cowell, R. G., Dawid, A. P., Lauritzen, S. L. and Spiegelhalter, D. J. (1999)
Probabilistic Net-works and Expert Systems: Exact Computational Methods for Bayesian Networks , vol. 1. 175,Fifth Avenue, New York, 10010, USA: Springer. An optional note.Dean, T. and Kanazawa, K. (1989) A model for reasoning about persistence and causation.
Computational Intelligence , , 142 – 150.FAO (1996) Rome declaration on world food security. Tech. rep. , Food and Agriculture Organi-sation of the United Nations.Faught, E., Williams, P., Willows, N., Asbridge, M. and Veugelers, P. (2017) The associationbetween food insecurity and academic achievement in canadian school-aged children.
PublicHealth Nutrition , , 1–8.Field, F., Thornton, T., Glen, J., Jenkin, B., Lewell-Buck, E. and Newton, S. (2014) Feedingbritain: A strategy for zero hunger in england, wales, scotland and northern ireland. the reportof the all-party parliamentary inquiry into hunger in the united kingdom. Tech. rep. , All-PartyParliamentary Inquiry into Hunger in the United Kingdom.Friel, S. and Ford, L. (2015) Systems, food security and human health.
Food Security , , 437–451.URL: http://dx.doi.org/10.1007/s12571-015-0433-1 .Fuller, E., Bankiewicz, U., Davies, B., Mandalia, D. and Stocker, B. (2019) The food and yousurvey wave 5 combined report for england, wales and northern ireland. Tech. rep. , FoodStandards Agency UK.Garratt, E. (2015) Csi 13: Food insecurity and foodbank use.
Tech. rep. , Centre for SocialInvestigations, Oxford.Hill, K. (2014) Measuring disadvantaged pupils attainment gaps over time: working methodologyand statistics for 2012-2014. Working paper SFR 40/2014, Department for Education.
Description of variables used in the network Keeney, R. L. and Raiffa, H. (1993)
Decision with Multiple Objectives: Preferences and ValueTrade-offs . Cambridge University Press.Kennedy, G., Berardo, A., Papavero, C., Horjus, P., Ballard, T., Dop, M., Delbaere, J. andBrouwer, I. D. (2012) Proxy measures of household food consumption for food security assess-ment and surveillance: Comparison of the household dietary diversity and food consumptionscores.
Public Health Nutrition , , 2010–2018.Kirkpatrick, S. and Tarasuk, V. (2009) Food insecurity and participation in community foodprograms among low-income toronto families. Can J Public Health , Mar-Apr , 35–39.Lagi, M., Bar-Yam, Y., Bertrand, K. and Bar-Yam, Y. (2012) Update february 2012. the foodcrises: A quantitative model of food prices including speculators and ethanol conversion. arXiv:1203.1313 , . , .Lee, A. M., Scharf, R. J., Filipp, S. L., Gurka, M. J. and DeBoer, M. D. (2019) Food Inse-curity Is Associated with Prediabetes Risk Among U.S. Adolescents, NHANES 2003–2014. METABOLIC SYNDROME AND RELATED DISORDERS , , 347–354.Leonelli, M. and Smith, J. Q. (2015) Bayesian decision support for complex systems with manydistributed experts. Annals of Operations Research , 1–26. URL: http://dx.doi.org/10.1007/s10479-015-1957-7 .Loopstra, R. (2014)
Household Food Insecurity in Canada: Towards and understanding of Effec-tive Interventions . Ph.D. thesis, University of Toronto.Loopstra, R., Reeves, A. and Stuckler, D. (2015a) Rising food insecurity in europe.
The Lancet , , 2041.Loopstra, R., Reeves, A., Taylor-Robinson, D., Barr, B., McKee, M. and Stuckler, D. (2015b)Austerity, sanctions, and the rise of food banks in the uk. BMJ , .Pan, L., Sherry, B., Njai, R. and Blanck, H. M. (2012) Food insecurity is associated with obesityamong us adults in 12 states. Journal of the Academy of Nutrition and Dietetics , , 1403–1409.Phillips, L. D. (1984) A theory of requisite decision models. Acta Psychologica , , 29–48.Pilgrim, A., Barker, M., Jackson, A., Ntani, G., Crozier, S., Inskip, H., Godfrey, K., Cooper, C.,S, R. and Group., S. S. (2012) Does living in a food insecure household impact on the dietsand body composition of young children? findings from the southampton women’s survey. JEpidemiol Community Health. , , .Queen, C. M. and Smith, J. Q. (1993) Multiregression dynamic models. Journal of the RoyalStatistical Society. Series B (Methodological) , , 849–870.Robert, C. and Casella, G. (2004) Monte Carlo Statistical Methods . Springer.Seligman, H. K., Laraia, B. A. and Kushel, M. B. (2010) Food insecurity is associated withchronic disease among low-income nhanes participants.
The Journal of Nutrition , , 304–310.Semega, J. L., Kollar, M. A., Creamer, J. and Mohanty, A. (2019) Income and poverty in theunited states: 2018. Tech. rep. , United States Census Bureau.
Description of variables used in the network Smith, J. Q. (2010)
Bayesian Decision Analysis: Principles and Practice . Cambridge UniversityPress.Smith, J. Q., Barons, M. and Leonelli, M. (2016) Coherent inference for integrating decisionsupport systems. arXiv . Http://arxiv.org/abs/1507.07394.Smith, J. Q., Barons, M. J. and Leonelli, M. (2015) Decision focused inference on networkedprobabilistic systems: with applications to food security. 3220–3233.Smith, J. Q., Faria, A. E., French, S., Ranyard, D., Vlesshhouwer, D., Bohunova, J., Duranova,T., Stubna, M., Dutton, L., Rojas, C. and Sohier, A. (1997) Probabilistic data assimilationwithin RODOS.
Radiation Protection Dosimetry , , 57–59.StatCan (2017) Canadian income survey, 2017. Tech. rep. , StatCan.Tarasuk, V., Fitzpatrick, S. and Ward, H. (2010) Nutrition inequities in canada.
Applied Physi-ology, Nutrition, and Metabolism. , , 172–179.Tarasuk, V., Mitchell, A. and Dachner, N. (2016) Household food insecurity in canada,2014. Tech. rep. , Toronto: Research to identify policy options to reduce food in-security (PROOF). URL: http://proof.utoronto.ca/wp-content/uploads/2016/04/Household-Food-Insecurity-in-Canada-2014.pdf .Taylor-Robinson, D., Rougeaux, E., Harrison, D., Whitehead, M., Barr, B. and Pearce, A. (2013)The rise of food poverty in the uk.
BMJ , , .Tingay, R. S., Tan, C. J., Tan, N. C., Tang, S., Teoh, P. F., Wong, R. and Gulliford, M. C.(2003) Food insecurity and low income in an english inner city. Journal of Public Health , ,156–159. URL: http://jpubhealth.oxfordjournals.org/content/25/2/156.abstract .Trust, T. (2019) End of year stats april 2018 – march 2019. Tech. rep. , Trussell Trust.United Nations Office of the High Commissioner (1966) International covenant on economic, so-cial and cultural rights. , Last accessed on 11/02/2020.USDA (2012) U.s. adult food security survey module.
Tech. rep. , USDA.West, M. and Harrison, J. (1997)