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ST3189 Machine Learning

创建日期：2024-05-09 15:09:06

所属分类：计算机computer science

标签： ST3189

Machine Learning

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ST3189

Machine Learning
Suitable for all candidates
Instructions to candidates
This paper contains four questions. Answer ALL FOUR. All questions will be given
equal weight (25%).
The marks in brackets reflect marks for each question.
Time allowed - Reading Time: None
Writing Time: 2 hours
You are supplied with: Graph paper
You may also use: No additional materials
Calculators: Calculators are allowed in this examination
1. (a) Suppose that yi ∼ N(µ, 1) for i = 1, . . . , n and that the yi’s are independent.
i. Show that the sample mean estimator µˆ1 =

which can be recognised as the IGamma(100.1, 20.1) distribution.
(b) Find the Jeffreys’ prior for λ. Which is the corresponding posterior distribu-
tion? [6 marks]

Hence Jeffreys’ prior is pi(λ) ∝ I(λ)1/2 ∝ (λ−2)1/2 = λ−1. The posterior
becomes
pi(θ|x) ∝ λ−100 exp
(
−20
λ
)
λ−1 = λ−100−1 exp
(
−20
λ
)
which can be recognised as the IGamma(100, 20)
(c) Find a Bayes estimator for λ based on the priors of parts (a) and (b). [3 marks]
Answer: A standard Bayes estimator is the posterior mean which is equal to
(see appendix)
20.1
100.1− 1 = 0.203
or
20
100− 1 = 0.202
depending on the chosen prior.
4
(d) Let y represent a future observation from the same model. Find the predictive
distribution of y based either on the prior of part (a) or (b). [6 marks]

(e) Describe how you can calculate the mean the of the predictive distribution in
software such as R. [5 marks].
Answer: Note that we can write the mean of the predictive distribution as
E(y|x) =
∫ ∞
0
yf(y|λ)pi(λ|x)dλ.
Hence a Monte Carlo scheme would draw samples y{k}, k = 1, . . . , N from
f(y|λ)pi(λ|x) for some large N and then just take
Ê(y|x) =
∑N
k=1 y
{k}
N
,
To that in R once can
i. Draw -say- 10,000 samples from the Gamma(100,20) using nu=rgamma(100,20)
.
ii. Invert those samples to make them samples from the IGamma(100,20)
using lambda=1/nu.
iii. Using each of the samples in lambda, sample y by the model y ∼Exponential(lambda)
using y=rexp(lambda).
iv. Calculate the sample mean of the values in y using mean(y).
5
3. (a) i. Suppose a non-linear model that can be written as
Y = f(X) + ,
where has zero mean and variance σ2, and is independent of X. Show
that the expected test error, conditional on X can be decomposed into the
following three parts:
E
[(
Y − fˆ(X)
)2]
= σ2 + Bias [f(x)]2 + Var [f(x)] ,
where f(·) is estimated from the training data. [7 marks]

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MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 EARN 5 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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 MGMTMFE 405 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