This exercise derives the normalization constant of Beta(a, 3) in the case of integer parameters a = s +1,3=n-s+1 by exploring the connection between • Bayesian inference for a Bernoulli parameter using uniform prior, and • Order statistics of a uniform RV. Let p = [0, 1] be the parameter of a Bernoulli(p) distribution (e.g. the probability of Heads for a coin). Suppose we have no prior information about p. In the Bayesian approach, we model our ignorance by considering the parameter p as a uniformly distributed random variable p~ Uniform([0, 1]). We can then model the observations X₁, X2,, Xn in the following way: let U₁, U2, Un be i.i.d. ~ Uniform([0, 1]) that are independent from p, and define (ii) Deduce that X₂ = 1U₁ p. P(X₁ + X₂ + + X₂ = s|p) = = 3 be the order statistics of U₁1, U2, =p³(1-p)"-s (") ₁² (1 P(X₁ + X₂ + + X₂ = 8) = (Hint: for the second equation, use Law of Total Probability.) p³ (1 - p)n-s p³ (1 - p)"-sdp. Y₁ ≤ Y₂ ≤ ≤ Yn+1 Un+1. Reason that P(Ys+1 = p) = P(X₁ + X₂ + + Xn = s).
This exercise derives the normalization constant of Beta(a, 3) in the case of integer parameters a = s +1,3=n-s+1 by exploring the connection between • Bayesian inference for a Bernoulli parameter using uniform prior, and • Order statistics of a uniform RV. Let p = [0, 1] be the parameter of a Bernoulli(p) distribution (e.g. the probability of Heads for a coin). Suppose we have no prior information about p. In the Bayesian approach, we model our ignorance by considering the parameter p as a uniformly distributed random variable p~ Uniform([0, 1]). We can then model the observations X₁, X2,, Xn in the following way: let U₁, U2, Un be i.i.d. ~ Uniform([0, 1]) that are independent from p, and define (ii) Deduce that X₂ = 1U₁ p. P(X₁ + X₂ + + X₂ = s|p) = = 3 be the order statistics of U₁1, U2, =p³(1-p)"-s (") ₁² (1 P(X₁ + X₂ + + X₂ = 8) = (Hint: for the second equation, use Law of Total Probability.) p³ (1 - p)n-s p³ (1 - p)"-sdp. Y₁ ≤ Y₂ ≤ ≤ Yn+1 Un+1. Reason that P(Ys+1 = p) = P(X₁ + X₂ + + Xn = s).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Exercise 6
This exercise derives the normalization constant of Beta(a, 3) in the case of integer
parameters as+1, B=n-s+ 1 by exploring the connection between
• Bayesian inference for a Bernoulli parameter using uniform prior, and
Order statistics of a uniform RV.
Let p = [0, 1] be the parameter of a Bernoulli(p) distribution (e.g. the probability of Heads
for a coin). Suppose we have no prior information about p. In the Bayesian approach,
we model our ignorance by considering the parameter p as a uniformly distributed random
variable
p~ Uniform ([0, 1]).
We can then model the observations X₁, X2,, Xn in the following way: let U₁, U2,, Un
be i.i.d. ~ Uniform([0, 1]) that are independent from p, and define
X₁ = 1u₁<p :=
(i) Reason that conditioned on the value of p, X₁, X2, Xn are i.i.d. ~ Bernoulli(p).
Conclude that
where s ₁ + x₂ + ··· + xn.
(ii) Deduce that.
and
P(X₁ = x₁, , Xn = n/p) = [p²¹ (1 - p)¹-ªi
i=1
P(X₁ + X₂ + + Xn=&
if U₁ ≤ P,
if U₂ > p.
(iii) Define Un+1 = p, and let
(iv) Reason that p =
3
n
=p³ (1 - p)"
sp) =
P(X₁ + X₂ + + Xn
· L² (®)»^«²-²
(Hint: for the second equation, use Law of Total Probability.)
s) =
- (") ₁² (1
³(1 − p)¹–s
Y₁ ≤ Y₂ ≤ ≤ Yn+1
be the order statistics of U₁, U2,,Un+1. Reason that
n-8
(v) Combine the previous parts to conclude that
p³(1 − p)"-³dp =
P(Ys+1 = p) = P(X₁ + X₂ + ··· + Xn = s).
1
n+1'
Yk with k = 1, 2,..., n + 1 equally likely. Conclude that
P(Ys+1 =P)
- p)n-sdp.
s! (ns)!
(n + 1)!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00bea801-7d41-45bf-88ce-69b6fd07ca20%2F0330c805-a09f-4bff-8648-27bfb8fe128e%2Fuigwyn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 6
This exercise derives the normalization constant of Beta(a, 3) in the case of integer
parameters as+1, B=n-s+ 1 by exploring the connection between
• Bayesian inference for a Bernoulli parameter using uniform prior, and
Order statistics of a uniform RV.
Let p = [0, 1] be the parameter of a Bernoulli(p) distribution (e.g. the probability of Heads
for a coin). Suppose we have no prior information about p. In the Bayesian approach,
we model our ignorance by considering the parameter p as a uniformly distributed random
variable
p~ Uniform ([0, 1]).
We can then model the observations X₁, X2,, Xn in the following way: let U₁, U2,, Un
be i.i.d. ~ Uniform([0, 1]) that are independent from p, and define
X₁ = 1u₁<p :=
(i) Reason that conditioned on the value of p, X₁, X2, Xn are i.i.d. ~ Bernoulli(p).
Conclude that
where s ₁ + x₂ + ··· + xn.
(ii) Deduce that.
and
P(X₁ = x₁, , Xn = n/p) = [p²¹ (1 - p)¹-ªi
i=1
P(X₁ + X₂ + + Xn=&
if U₁ ≤ P,
if U₂ > p.
(iii) Define Un+1 = p, and let
(iv) Reason that p =
3
n
=p³ (1 - p)"
sp) =
P(X₁ + X₂ + + Xn
· L² (®)»^«²-²
(Hint: for the second equation, use Law of Total Probability.)
s) =
- (") ₁² (1
³(1 − p)¹–s
Y₁ ≤ Y₂ ≤ ≤ Yn+1
be the order statistics of U₁, U2,,Un+1. Reason that
n-8
(v) Combine the previous parts to conclude that
p³(1 − p)"-³dp =
P(Ys+1 = p) = P(X₁ + X₂ + ··· + Xn = s).
1
n+1'
Yk with k = 1, 2,..., n + 1 equally likely. Conclude that
P(Ys+1 =P)
- p)n-sdp.
s! (ns)!
(n + 1)!
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