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COMP3670: Introduction to Machine Learning

创建日期：2024-08-05 15:05:47

所属分类：计算机computer science

标签： COMP3670

Introduction to Machine Learning

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COMP3670: Introduction to Machine Learning

Errata: All corrections are in red.

Note: For the purposes of this assignment, if X is a random variable we let pX denote the probability

density function (pdf) of X, FX to denote it’s cumulative distribution function, and P to denote

probabilities. These can all be related as follows:

P (X ≤ x) = FX(x) =∫ x∞pX(z)dz

P (a ≤ X ≤ b) = FX(b)? FX(a) =∫ bapX(z)dz

Often, we will simply write pX as p, where it’s clear what random variable the distribution refers to.

You should show your derivations, but you may use a computer algebra system (CAS) to assist

with integration or differentiation. We are not assessing your ability to integrate/differentiate here.1.

Question 1 Continuous Bayesian Inference 5+5+2+4+4+6+6+5=37 credits

Let X be a random variable representing the outcome of a biased coin with possible outcomes X =

{0, 1}, x ∈ X . The bias of the coin is itself controlled by a random variable Θ, with outcomes2 θ ∈ θ,

where

θ = {θ ∈ R : 0 ≤ θ ≤ 1}

The two random variables are related by the following conditional probability distribution function of

X given Θ.

p(X = 1 | Θ = θ) = θ

p(X = 0 | Θ = θ) = 1? θ

We can use p(X = 1 | θ) as a shorthand for p(X = 1 | Θ = θ).

We wish to learn what θ is, based on experiments by flipping the coin.

We flip the coin a number of times.3 After each coin flip, we update the probability distribution for θ

to reflect our new belief of the distribution on θ, based on evidence.

Suppose we flip the coin n times, and obtain the sequence of coin flips 4 x1:n.

a) Compute the new PDF for θ after having observed n consecutive ones (that is, x1:n is a sequence

where ?i.xi = 1), for an arbitrary prior pdf p(θ). Simplify your answer as much as possible.

b) Compute the new PDF for θ after having observed n consecutive zeros, (that is, x1:n is a sequence

where ?i.xi = 0) for an arbitrary prior pdf p(θ). Simplify your answer as much as possible.

c) Compute p(θ|x1:n = 1n) for the uniform prior p(θ) = 1.

d) Compute the expected value μn of θ after observing n consecutive ones, with a uniform prior

p(θ) = 1. Provide intuition explaining the behaviour of μn as n→∞.

1For example, asserting that

∫ 10x2(x3 + 2x)dx = 2/3 with no working out is adequate, as you could just plug the

integral into Wolfram Alpha using the command Integrate[x^2(x^3 + 2x),{x,0,1}]

2For example, a value of θ = 1 represents a coin with 1 on both sides. A value of θ = 0 represnts a coin with 0 on

both sides, and θ = 1/2 represents a fair, unbaised coin.

3The coin flips are independent and identically distributed (i.i.d).

4We write x1:n as shorthand for the sequence x1x2 . . . xn.

e) Compute the variance σ2n of the distribution of θ after observing n consecutive ones, with a uniform

prior p(θ) = 1. Provide intuition explaining the behaviour of σ2n as n→∞.

f) Compute the maximum a posteriori estimation θMAPn of the distribution on θ after observing

n consecutive ones, with a uniform prior p(θ) = 1. Provide intuition explaining how θMAPn varies

with n.

g) Given we have observed n consecutive coin flips of ones in a row, what do you think would be a

better choice for the best guess of the true value of θ? μn or θMAP ? Justify your answer. (Assume

p(θ) = 1.)

h) Plot the probability distributions p(θ|x1:n = 1) over the interval 0 ≤ θ ≤ 1 for n ∈ {0, 1, 2, 3, 4} to

compare them. Assume p(θ) = 1.

Question 2 Bayesian Inference on Imperfect Information (4+5+8+4+4=25 credits)

We have a Bayesian agent running on a computer, trying to learn information about what the pa-

rameter θ could be in the coin flip problem, based on observations through a noisy camera. The noisy

camera takes a photo of each coin flip and reports back if the result was a 0 or a 1. Unfortunately, the

camera is not perfect, and sometimes reports the wrong value.5 The probability that the camera makes

mistakes is controlled by two parameters α and β, that control the likelihood of correctly reporting a

zero, and a one, respectively. Letting X denote the true outcome of the coin, and X denoting what

the camera reported back, we can draw the relationship between X and X? as shown.

We would now like to investigate what posterior distributions are obtained, as a function of the

parameters α and β.

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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 EMESTER1 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