Assignment 4 Question 1 Continuous Bayesian Inference


Rate this product

Research School of Computer Science Assignment 4 Theory Questions
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) = Z x
P(a ≤ X ≤ b) = FX(b) − FX(a) = Z b
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,
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 R 1

3 + 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 σ
n of the distribution of θ after observing n consecutive ones, with a uniform
prior p(θ) = 1. Provide intuition explaining the behaviour of σ
n 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 parameter 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 Xb denoting what
the camera reported back, we can draw the relationship between X and Xb as shown.
X = 0 Xb = 0
X = 1 Xb = 1
1 − θ
1 − α
1 − β
So, we have
p(Xb = 0 | X = 0) = α
p(Xb = 0 | X = 1) = 1 − β
p(Xb = 1 | X = 1) = β
p(Xb = 1 | X = 0) = 1 − α
We would now like to investigate what posterior distributions are obtained, as a function of the
parameters α and β.
a) (5 credits) Briefly comment about how the camera behaves for α = β = 1, for α = β = 1/2, and
for α = β = 0. For each of these cases, how would you expect this would change how the agent
updates it’s prior to a posterior on θ, given an observation of Xb? (No equations required.) You
shouldn’t need any assumptions about p(θ) for this question.
b) (10 credits) Compute p(Xb = x|θ) for all x ∈ {0, 1}.
5The errors made by the camera are i.i.d, in that past camera outputs do not affect future camera outputs.
c) (15 credits) The coin is flipped, and the camera reports seeing a one. (i.e. that Xˆ = 1.)
Given an arbitrary prior p(θ), compute the posterior p(θ|Xˆ = 1). What does p(θ|Xˆ = 1) simplify
to when α = β = 1? When α = β = 1/2? When α = β = 0? Explain your observations.
d) Compute p(θ|Xˆ = 1) for the uniform prior p(θ) = 1. Simplify it under the assumption that β := α.
e) (10 credits) Let β = α. Plot p(θ|Xˆ = 1) as a function of θ, for all α ∈ {0,
, 1} on the same
graph to compare them. Comment on how the shape of the distribution changes with α. Explain
your observations. (Assume p(θ) = 1.)
Question 3 Relating Random Variables (10+7+5+16=38 credits)
A casino offers a new game. Let X ∼ fX be a random variable on (0, 1] with pdf pX. Let Y be a
random variable on [1, ∞) such that Y = 1/X. A random number c is sampled from Y , and the player
guesses a number m ∈ [1, ∞). If the player’s guess m was lower than c, then the player wins m − 1
dollars from the casino (which means higher guesses pay out more money). But if the player guessed
too high, (m ≥ c), they go bust, and have to pay the casino 1 dollar.
a) Show that the probability density function pY for Y is given by
pY (y) = 1
b) Hence, or otherwise, compute the expected profit for the player under this game. Your answer will
be in terms of m and pX, and should be as simplified as possible.
c) Suppose the casino chooses a uniform distribution over (0, 1] for X, that is,
pX(x) = (
1 0 < x ≤ 1
0 otherwise
What strategy should the player use to maximise their expected profit?
d) Find a pdf pX : (0, 1] → R such that for any B > 0, there exists a corresponding player guess m
such that the expected profit for the player is at least B. (That is, prove that the expected profit
for pX, as a function of m, is unbounded.)
Make sure that your choice for pX is a valid pdf, i.e. it should satisfy
Z 1
pX(x)dx = 1 and pX(x) ≥ 0
You should also briefly mention how you came up with your choice for pX.
Hint: We want X to be extremely biased towards small values, so that Y is likely to be large, and
the player can choose higher values of m without going bust.


There are no reviews yet.

Be the first to review “Assignment 4 Question 1 Continuous Bayesian Inference”

Your email address will not be published. Required fields are marked *

Scroll to Top