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1 Bayesian Risk and Decision Analysis Semester B, 2020 Coursework 2: “Too Good to be True” Instructions This coursework requires an answer in two parts. Your answer should be prepared using the MSWord word processor and submitted electronically. All calculation/analysis steps must be clearly shown, and any formal mathematical working shown explicitly. This coursework is marked out of 100 and counts for 15% of the final mark for the module. This is an individual coursework. ALL submissions will be carefully screened for signs of group work (including similar screen dumps and formatting) and if found College regulations governing assessment will apply. Questions “If everyone is thinking alike, then no one is thinking” Benjamin Franklin This coursework is inspired by an example from this Royal Society paper: https://royalsocietypublishing.org/doi/10.1098/rspa.2015.0748 According to ancient Jewish law if a panel of the Sanhedrin (23 Jewish judges) unanimously find a defendant guilty of murder then they must acquit on the assumption that their verdict is most likely to be “too good to be true”. The underlying assumption behind this rule is that, in this situation, the Sanhedrin’s judgement may be systematically biased, and hence the judgements may be dependant, one on another, rather than each judgement produced independently. Question 1: Use AgenaRisk to build a Bayesian Network model to represent this situation using this information: There are up to ? =22 judges ? = 0,1,2, … … . ? on a panel of judges. The actual guilt of a suspect is a Boolean node, ??????? ?????. The verdict pronounced by a judge is a Boolean node, ??????? ?????. All judges biased or not is a Boolean node, ??????. https://royalsocietypublishing.org/doi/10.1098/rspa.2015.0748 2 ?(??????? ????? = ????) = 0.4 ? ?(??????? ????? = ???? | ??????? ????? = ????, ?????? = ?????) = 0.7 ? ?(??????? ????? = ???? | ??????? ????? = ?????, ?????? = ?????) = 0.3 ? ?(??????? ????? = ???? | ??????? ????? = ?????, ?????? = ????) = 0.8 ? ?(??????? ????? = ???? | ??????? ????? = ????, ?????? = ????) = 0.8 ?(?????? = ????) = 0.1 a) Show a screen shot of your BN identifying the relevant variables to model this problem, where the Sanhedrin can be independent or dependent (biased). [15 marks] b) Show the conditional probability tables for all relevant variables. [15 marks] Hint: The way they present the model in the Royal Society paper may not be the way you might choose to follow. For instance, you do not necessarily need to use a binomial distribution but can instead simply build a discrete BN with separate Bernoulli trial nodes for each judge. However, whatever method you choose to follow any correct method is equally acceptable. Note: The model used in the paper is not identical to the model required here. You might want to build the model in the paper first to verify your results and then amend your parameters to match those required for this coursework. Question 2: Create seven scenarios (the ???????? ????) in AgenaRisk and execute a calculation where each of the seven scenarios represents the total number of judges sitting, who pronounce guilty verdicts: • 1 judge total with one guilty verdict (22 unobserved/ “no answer”) • 2 judges total with two guilty verdicts (21 unobserved/ “no answer”) • 5 judges total with five guilty verdicts (18 unobserved/ “no answer”) • 10 judges total with ten guilty verdicts (13 unobserved/ “no answer”) • 15 judges total with fifteen guilty verdicts (8 unobserved/ “no answer”) • 20 judges total with twenty guilty verdicts (3 unobserved/ “no answer”) • 23 judges total with twenty-three guilty verdicts (0 unobserved/ “no answer”) Show a screenshot of the ‘Risk Table’ in AgenaRisk showing the observations entered in all scenarios (Look in the AgenaRisk manual for information on the Risk Table). [10 marks] a) Show screenshots of AgenaRisk showing the posterior marginal distributions for ?(??????? ????? = ???? | ???????? ????) and ?(??? ?????? ?????? = ???? | ???????? ????) for all scenarios. [20 marks] b) Take the posterior marginals for ?(??????? ????? = ???? | ???????? ????) and ?(??? ?????? ?????? = ???? | ???????? ????) and plot, using Excel or similar, the 3 response curve with x-axis equal to the number of judges in each scenario and y-axis equal to these posterior marginals. [20 marks] c) From your answer to Question c): i. When is the posterior belief in Guilt at its maximum? [5 marks] ii. When is the posterior belief in Bias greater than the posterior belief in Guilt? [5 marks] Note: Provide approximate values or ranges for these values. d) Explain, in your own words, the results from Q2 c) and explain why the verdicts of the judges may not be independent. [5 marks] e) Identify an area of scientific, social or political discourse, where there is near unanimity of opinion but where a suspicion of systematic bias might be reasonably entertained. [5 marks]