MATH3871/MATH5960 Bayesian Inference and Computation Lab 1 Exercises These exercises provide some practice in performing basic Bayesian analyses using R. There is no requirement to do the exercises in order. Outline solutions are available in a separate file. 1) Water consumption Water consumption Water use More than 5 million ML of water is used in Victoria each year, 90% from surface water and 10% from groundwater. The majority of Victoria’s water resource is used for irrigation (78%), while urban uses (both metropolitan and regional) account for 17% of Victoria’s water consumption. Efficiency of water use2 Making the best use of our water resource CSIRO has provided a perspective on water use which is related to the contribution it makes to Gross National Expenditure (GNE). The prices we pay for agricultural products tend not to take into account the full environmental costs of production. When water inputs are considered there is variation in the amount of water used to produce various commodities. For example, rice production uses more than 8,000 litres of water for every dollar of GNE, compared to cereal production which uses around 600 litres to produce the same value of product.2 Trends in water consumption While time-series water consumption data are not available for regional Victoria, records for Melbourne show that per capita water consumption grew steadily between the 1940s and 1960s, with strong increases in the 1970s. Since the 1980s water consumption has been influenced by drought, and associated water restrictions, as well as by conservation and efficiency incentives and market reforms designed to reduce water consumption over the longer term.3 Average daily per capita water use4 Melbourne 1940-2004* * NOTE: Figure for 2003-04 is forecasted estimation Consumptive uses of water in Victoria1 2003-04 BTRc^a ;XcaTb^UfPcTa_Ta^U6a^bb=PcX^]P[4g_T]SXcdaT AXRT 2^cc^] 3PXah
BdVPaRP]T 3PXah_a^SdRcb 1TTU 5[^daRTaTP[U^^Sb BTaeXRTbc^PVaXRd[cdaT '#' % # #$! !#& '$! &" % ' $'# Sources 1DSE 2005 State Water Report 2CSIRO and University of Sydney 2005 Balancing Act. A Triple Bottom Line Analysis of the Australian Economy 3Victorian Government 2004 Securing Our Water Future Together 4Melbourne Water 2005 A Dry History & 8aaXVPcX^] AdaP[3^\TbcXRP]SBc^RZ ATVX^]P[DaQP] &' $ 1 9 4 0 1 9 4 2 1 9 4 4 1 9 4 6 1 9 4 8 1 9 5 0 1 9 5 2 1 9 5 4 1 9 5 6 1 9 5 8 1 9 6 0 1 9 6 2 1 9 6 4 1 9 6 6 1 9 6 8 1 9 7 0 1 9 7 2 1 9 7 4 1 9 7 6 1 9 7 8 1 9 8 0 1 9 8 2 1 9 8 4 1 9 8 6 1 9 8 8 1 9 9 0 1 9 9 2 1 9 9 4 1 9 9 6 1 9 9 8 2 0 0 0 2 0 0 2 2 0 0 4 Financial Year Ending L i t r e s p e r p e r s o n p e r d a y 550 500 450 400 350 300 250 200 150 100 50 0 Drought Years Water Restrictions Mid 1990s: COAG Water Reforms - Pricing measures introduced 86 NOTE: These data are for Australia overall. Cotton is not grown in Victoria and rice represents less than 1% of the State’s grain production (as at 2001 Agricultural Census) In the Melbourne average daily per capita water use analyis, we modelled the discrete observa- tions x1, . . . , xn as independent draws from a Poisson(θ) distribution. Assuming a Gamma(α, β) prior, which has a density function of pi(θ) = βα Γ(α) θα−1 exp(−βθ), for α, β, γ > 0, we computed the posterior as a Gamma (α + ∑n i=1 xi, β + n) distribution. (a) Given that n = 65, ∑ i xi = 24, 890 and with prior parameters α = 1, β = 0.01, compute a point estimate (i.e. the posterior mean) and a 95% central credible interval for θ. Note, you will need to compute the credible interval numerically in R (hint: use the R command qgamma). (b) Draw a sample of sizeN = 500 directly from the posterior distribution (see the R command rgamma), and obtain Monte Carlo estimates of the lower and upper values of the 95% credible interval for θ. Repeat this 250 times, and produce a histogram for the distribution of each interval end- point. Superimpose a point corresponding to the true interval endpoints. How accurate is the Monte Carlo estimate? (R commands: hist, points). 1 Produce another pair of histograms, but this time use N = 5000 samples. How is the precision of the Monte Carlo estimates affected? How many samples, N , are needed for the spread (i.e. min - max) of the Monte Carlo estimates for each interval endpoint to be less than 0.15? (c) In lectures it was stated that the predictive distribution for a future observation, y, is NegBin ( y | α +∑i xi, 1β+n+1). Draw samples directly from the posterior distribution. Use these to obtain samples from the posterior predictive distribution, and plot this via a histogram. Superimpose the density of the algebraically computed negative binomial predictive distribution (R command: dnbinom). Do the distributions coincide? (d) What are the advantages/disadvantages of performing statistical analyses using the alge- braically exact approach, and the Monte Carlo approximations? 2) Estimating evolutionary fitness of tuberculosis Untreated, Sensitive Untreated, Resistant Treated, Resistant (acquired by mutation) Treated, Resistant (via transmission) Treated, Sensitive c c c S R R Luciani et al. (2009) developed a stochastic model (above) to estimate epidemiological param- eters relating to the development of drug resistance in mycobacterium tuberculosis. Unknown model parameters included the transmission rate (α), the marker mutation rate (µ), the mu- tation rate of drug resistance (ρ) and the transmission cost due to resistance (c). The rates of cure due to treatment for sensitive (εS) and resistant (εR) strains, and the detection and treatment rate (τ) are held fixed. Samples from the posterior distribution when analysing the IS6110 marker from Cuban data can be found in the file tuberculosis.txt (the rows correspond to α, c, ρ and µ in order). (a) Read the posterior into R (using the command read.table). Produce marginal posterior histograms of each parameter, and scatterplots of the 6 bivariate distributions (e.g. (α, c), . . . , (ρ, µ)). (b) The relative fitness of the drug-resistant strains based on the model of Luciani et a. (2009) can be expressed as Φ = (1− c) 1 τ + 1 δ+εR 1 τ + 1 δ+εS+ρ . 2 If δ = τ = εS = 0.52 and εR = 0.202 are fixed, then produce a histogram of the posterior distribution of Φ. Calculate Pr(Φ < 1), the posterior probability that Φ < 1. Is there any evidence that the resistant strain is any less evolutionarily fit than the susceptible strain? (i.e. is there any evidence that Φ < 1?) (c) A central 95% credible interval for ρ can be estimated by discarding the lower and upper 2.5% of the posterior samples. Obtain such a 95% interval for ρ and comment on its length. (d) A 95% credible interval is any interval for which the interval contains 95% of the poste- rior density. For example, (q0.01, q0.96), where qx is the x-th quantile of a parameter, is also a 95% credible interval. As there are (in theory) an infinite number of 95% credible intervals for any parameter, convention a useful strategy is to take the shortest one. Based on the posterior sample of length 5000, compute 250 unique 95% credible intervals for ρ, and identify the shortest. How does this compare to using the central 95% credible interval? Under what circumstances is the central 95% credible interval likely to be close to the shortest length? 3) Buffon’s Needle One of the most famous simulation experiments is Buffon’s Needle, designed to calculate (not very efficiently!) an estimate of pi. Imagine a grid of parallel lines with spacing d, on which a needle of length ` ≤ d is dropped. We repeat this experiment n times and count the proportion of times, pˆ, that the needle intersects with a line. The rationale behind this is that if x is the distance from the centre of the needle to the leftmost line, and if θ is the angle from the vertical, then under the assumption of random needle throwing, we would have x ∼ U(0, d), and θ ∼ U(0, pi). Hence p = Pr(needle intersects line) = 1 pi ∫ pi 0 Pr(needle intersects|θ = φ)dφ = 1 pi ∫ ( 2 d × ` 2 sinφ ) dφ = 2` pid . 3 Hence, an estimate of pi is pˆi = 2` pˆd . (a) Produce some code to simulate the Buffon’s Needle experiment, given the lengths ` and d, and produce an estimate of pi. Plot the estimate of pi as the number of simulations, n, increases. (b) A natural question is how to optimise the relative sizes of ` and d. Consider the variability of 1/pˆi. Now npˆ ∼ Bin(n, p), so Var(pˆ) = p(1− p)/n. Show that Var(1/pˆi) = Var(pˆd/2`) = . . . = 1 npi2 ( pi 2ρ − 1 ) where ρ = `/d. When is this minimised (for 0 ≤ ρ ≤ 1)? (c) By computing the estimate of pi 1000 times and computing the standard deviation, for a range of values of ρ = `/d, empirically demonstrate that your optimal value of ρ leads to the smallest variability for pˆi. There are a number of things which may (or may not!) improve the efficiency of this experiment, including: • using a grid of rectangles or squares; • using a cross or other shape instead of a needle • using a needle of length greater than the grid separation. The point is: simulation can be used to answer many interesting problems, but careful design may be needed to achieve even moderate efficiency. 4 欢迎咨询51作业君
