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MAS2317 Presentation

MAS2317 Presentation. Thomas Smith. The Problem. (a) Obtaining the prior distribution for θ. Defining the limits of the distribution: The A&E department can handle at most 200 patients The least amount of patients there can be is zero

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MAS2317 Presentation

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  1. MAS2317 Presentation Thomas Smith

  2. The Problem

  3. (a) Obtaining the prior distribution for θ • Defining the limits of the distribution: • The A&E department can handle at most 200 patients • The least amount of patients there can be is zero • So using the match tool for the trial roulette method we set our limits We are told by the consultant that he has never seen more than 180 casualties in a day and has only on very few occasions seen less than 40 casualties a day. So we leave the bins below 40 and above 180 empty as the probability of these values occurring are negligible.

  4. (a) Obtaining the prior distribution for θ We are told to let Pr(80 < θ < 100) = a. We then elicit the following information from the consultant: So placing chips corresponding to the proportions the probabilities are given as to obtain the distribution: To make our prior distribution easier to use as we don’t need the parameters to such accuracy, we round the parameters. Thus our elicited prior distribution for θ is:

  5. (b) Is your prior in (a) conjugate To determine if the prior distribution is conjugate we need to see if the posterior distribution is from the same family of distributions as the prior distribution. So we need to calculate the posterior distribution. First we must calculate the likelihood function for the distribution of the number of admissions per day: We also have the prior distribution for θ:

  6. (b) Is your prior in (a) conjugate Now we must combine the likelihood function for the distribution of the number of admissions per day and the prior distribution for θusing Bayes’ theorem for distributions: Thus we get the posterior distribution: This shows the prior distribution is conjugate as the posterior and prior distributions are from the same family of distributions.

  7. (c) Produce a feedback summary for the A&E consultant • I was told to provide a feedback summary for the A&E consultant which gives: • The prior mode for θ • The prior standard deviation for θ • The 1% and 99% prior percentiles • Using the formulae given in the lecture notes the prior mode and standard deviation are: We also have the 1% and 99% prior percentiles which were given previously than the match elicitation tool:

  8. (c) Produce a feedback summary for the A&E consultant However this information could be better presented to the A&E consultant to make it easier for them to understand what we are showing them by producing a feedback summary as follows: “ From the information you have given me we have determined that the most common number of patients admitted in a day is about 82. The standard deviation of θ is 28.75 which is quite large therefore the number of patients admitted in a day will vary quite a lot and the values that occur will be quite spread out. We determined the smallest number of patients in a day that is not impossible but will only occur about 3 or 4 times a year is 38 and the greatest is 173. Do you believe this to be correct or do you think this should be modified? ”

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