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MATH3871/MATH5960 Assignment

创建日期：2024-05-15 14:59:23

所属分类：数学mathematics

标签： MATH3871

Assignment

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MATH3871/MATH5960

Assignment
This assignment covers material in Lectures from weeks 6-8. The Assignment is worth 15%
of the final course grade. 5% of the grade will be allocated for neat and concise presentation.
Please refer to the following instructions:
• Assignment to be submitted via Moodle by 14 November 11:55PM AEDT
• You may work individually OR in groups of max. 3 people. Each group should submit
only one submission (any member can upload the submission to moodle). All members
must sign the plagiarism statement and attach to the submission and clearly indicate who
is part of the group.
• Include in your assignment, any relevant R code and output and mathematical derivations.
Embed the code and plots into your assignment (please don’t attach R markdown or other
R script files)
• The total number of submitted pages should not exceed 7 A4 pages. Any pages submitted
in excess of 7 pages will not be graded.
• Print, sign and attach this cover sheet with your assignment (not included in page count).
• Refer to course handout for grading of late submissions
1
Plagiarism Statement
I declare that this assessment item is my own work, except where acknowledged, and has not
been submitted for academic credit elsewhere. I acknowledge that the assessor of this item may,
for the purpose of assessing this item reproduce this assessment item and provide a copy to
another member of UNSW; and/or communicate a copy of this assessment item to a plagiarism
checking service (which may then retain a copy of the assessment item on its database for the
purpose of future plagiarism checking).
I certify that I have read and understood UNSW Rules in respect of Student Academic Mis-
conduct.
Name (print clearly):
Student Number:
Signature:
Date:
2
Questions
1. Archaeology provides a rich source of complex, non-standard problems, where if the prior
is available, it needs careful elicitation. Here we look at a data study of the technique of
corbelling, a method of roofing spaces with blocks of stone, widely used in prehistory.
For many decades, archaeologists and historians have been fascinated by the ability of
prehistoric communities to develop sufficient skills to allow them to construct these domes,
some of which have survived for over 4,000 years. These speculations have led to applied
mathematical models being developed as an aid to understanding why the domes stand
up and how they were constructed.
Consider the simplest of these models
yi = αx
β
i
where y denotes the radius of the dome at which measurements were taken, and x is the
depth from the apex of the dome to the point at which measurements were taken. It is
easier to work with the log linear model
ln yi = lnα + β lnxi + i
where i ∼ N(0, σ2) are iid error terms.
Below are 24 measurements from the late Minoan tholos dome at Stylos, of Crete in
Greece.
x (depth) 0.04 0.24 0.44 0.64 0.84 1.04 1.24 1.44 1.64 1.84 2.04 2.24
y (radius) 0.40 0.53 0.70 0.90 1.06 1.16 1.26 1.36 1.47 1.62 1.67 1.68
x (depth) 2.44 2.64 2.84 3.04 3.24 3.44 3.64 3.84 4.04 4.24 4.44 4.64
y (radius) 1.77 1.82 1.89 1.96 2.00 2.05 2.10 2.10 2.14 2.13 2.15 2.14
Table 1: Measurements for the late Minoan tholos at Stylos.
Use the Metropolis-Hastings (M-H) algorithm to compute posterior estimates for the
parameters ln(α), β and σ2. Code your M-H sampler from scratch in R rather than using
in-built packages. In your answer, provide:
(a) specific settings of your M-H algorithm (e.g. chosen proposal q)
(b) The priors chosen, and their justification. Carry out a prior sensitivity analysis (i.e.
assess the sensitivity of your posterior distribution to moderate changes in your prior
specification).
(c) The likelihood function.
(d) Results of convergence assessments
(e) Traceplots, density plots of the posterior samples, and a suitable point estimate for
each posterior parameter.
(f) The R code developed (not included in page count)
3
2. Benford’s law, also called the first-digit law, states that in lists of numbers from many
(but not all) real-life sources of data, the leading digit 1 occurs much more often than
the others (about 30% of the time). Furthermore, the larger the digit, the less likely it is
to occur as the leading digit of a number. The approximate distribution of leading digits
(which exclude 0 by definition) is given by:
Probability
1 2 3 4 5 6 7 8 9
Leading Digit 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046
See http://en.wikipedia.org/wiki/Benford%27s law for more information on this
phenomenon.
Benford’s law can be used to expose cheating in tax and elections (among other things).
Below are counts of the number of polling stations (out of 192) that recorded each leading
digit for the number of pro-Hugo Chavez votes cast in a past Venezuelan election.
Number of Voting Stations
1 2 3 4 5 6 7 8 9
Leading Digit 31 32 29 20 18 18 21 13 10
(a) Assuming a multinomial likelihood and dirichlet prior with parameter vector a =
(a1, . . . , a9)
>, show that the Bayes factor for testing the conjecture that observed
counts n = (n1, . . . , n9)
> are consistent with Benford’s law is given by
B01 =
B(a)
B(a + n)
9∏
j=1
p
nj
0j
where p0 = (p01, . . . , p09)
> are Benford’s hypothesised proportions and
B(a) = B(a1, . . . , a9) =
∏9
j=1 Γ(aj)
Γ(
∑9
j=1 aj)
.
(b) Derive a similar expression for the fractional Bayes factor with training fraction b,
and produce a plot of 0 ≤ b ≤ 1 versus Bayes factor.
(c) Assess the adherance of the Venezuelation election counts to Benford’s law by eval-
uating both standard and fractional Bayes factors, and drawing appropriate conclu-
sions.
3. Let x1, . . . , xn be independent binary observations. Under model 1, Pr(xi = 1) = θ with
improper prior distribution pi(θ) = c1θ
−1(1 − θ)−1 for 0 < θ < 1, while under model 2,
Pr(xi = 1) = θ0, a fixed value. Let r =
∑n
i=1 xi be the number of ones in the data, and
suppose that 0 < r < n.
4
(a) Show that all minimal training samples produce the same partial Bayes factor, and
hence that both arithmetic (BA12) and geometric (B
G
12) intrinsic Bayes factors take
the value
BA12 = B
G
12 = B(r, n− r)θ1−r0 (1− θ0)1−n+r,
where B(p, q) = Γ(p)Γ(q)/Γ(p+ q) is the beta function.
(b) Derive the fractional Bayes factor BF12.
(c) Contrast BG12 and B
F
12 in the case θ0 = 0 and r = 1 by evaluating which models each
Bayes factor supports. Are these consistent with each other? Which is correct?

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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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