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STAT3016/6016 - Introduction to Bayesian Data Analysis
创建日期：2024-09-06 17:45:30
所属分类：经济Economics
标签：
Introduction to Bayesian Data Analysis

STAT3016/6016 - Introduction to Bayesian Data Analysis

Problem 1 [8 marks]

The Poisson-Gamma distribution is sometimes used as the sampling model assumption for observed count data that shows more dispersion than predicted under a Poisson model. Consider the data model PoissonGamma(a, b) (i = 1, ..., n). The probability density function of the Poisson-Gamma distribution is

where a > 0 and b > 0

Suppose that the parameters (a, b) are assigned the joint non-informative prior distribution proportional to 1/(ab)2 .

(a)  [1 mark] Is the joint non-informative prior distribution p(a, b) / 1/(ab)2                                                                       a

proper or improper prior distribution? Give a reason for your answer.

(b)  [1 mark] Suppose we consider transforming the parameters by taking the natural logarithm of a and b respectively. Define θ 1  = loge a and θ2  = loge b. Suggest one reason why such a transformation might be of interest.

(c)  [2 marks] Derive the prior density in terms of (θ1 , θ2). That is, findp(θ1 , θ2).  [Hint: you will need to apply the method of transformations technique for a bivariate density to derive the joint prior density of the transformed parameters]

(d)  [2 marks] Derive the joint posterior distribution p(θ1 , θ2 jy1, ..., yn) up to a proportionality constant (that is, ignoring any normalising constant).

(e)  [1 mark] The joint posterior density in part (d) is not a standard density that can be sampled directly from. What approximation approach could you use to obtain 1000 posterior draws of (θ1 , θ2 )?   (Note, you only need to provide the name of the approximation method, not describe how you would implement it)

(f)  [1  mark] Suppose you have generated 1000 posterior draws of (θ1 , θ2) from the target posterior density derived in part (d). Describe how you would use the simulated posterior sample to obtain 90% interval estimates for the parameters a and b.

Problem 2 [7 marks]

In the recent literature, the Poisson distribution has been used to model COVID-19 death counts.  Valid estimation of the death rate relies on the availability of accurate data on COVID-19 death counts.  In some regions in the world, the accuracy of reported COVID-19 death counts is questionable. We are going to build a Bayesian model to estimate the death rate parameter due to COVID-19 using a Poisson sampling model but also allow for possible under-reporting of death counts.

Let y1, ..., yn  be the reported daily death counts for Region A for n diferent days. Let z1 , ..., zn  be the true (unreported) death counts, the sampling model assumption is Pois(λ). (Note, in this question we assume there is no over-reporting of death counts, just under-reporting so yi ≤ zi.)

(a)  [2 marks] Assume yi  = zi   (i = 1, ..., n).  So there is no under-reporting of death counts. Define a prior distribution for λ and derive the posterior distribution of λ with your chosen prior.  (Note, you do not need to provide specific hyperparameter values in your chosen prior assumption).

(b)  [3 marks] Now we want to adjust our model for the possibility of under-reporting. The true rate of under-reporting is not known but it is assumed that the reported death count yi  is at least 80% of the true death count zi  (for i = 1, ..., n).  That is, the under-reporting rate is at most 20%.

Give the posterior distribution of λ up to a proportionality constant  (that is, ignoring any nor- malising constant), under this revised model

(c)  [2  marks]  Describe in words how we could use the posterior distribution in  (b) to  predict the unobserved true death counts zi.

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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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