Bayes Assignment 1 Draft

1st I will enter all 11 possible values for θ and the discrete prior distribution as given in the question. All values of θ are given to be equally likely hence p(θ)=1/11 for each of the 11 θ values

library(knitr)

## Warning: package 'knitr' was built under R version 4.4.3

library(ProbBayes)

## Warning: package 'ProbBayes' was built under R version 4.4.3

## Loading required package: LearnBayes

## Loading required package: ggplot2

## Warning: package 'ggplot2' was built under R version 4.4.3

## Loading required package: gridExtra

## Warning: package 'gridExtra' was built under R version 4.4.3

## Loading required package: shiny

## Warning: package 'shiny' was built under R version 4.4.3

bayes_table=data.frame(
  theta=seq(0,1,by=0.1),
  Prior=rep(1/11,11)# all equally likely
)

We know that the likelihood function p(y|θ) ~Binom(n,θ) , where our observed data is n=1000 and y=320. We use the dbinom() function to compute p(y|θ) for all θ values.

bayes_table$Likelihood=dbinom(320,1000,prob = bayes_table$theta)

We use the bayesian_crank function to calculate the posterior distribution p(θ|y) for all θ values by computing Bayes Rule.

#computing posterior
bayes_table=bayesian_crank(bayes_table)
kable(bayes_table) # prints out the bayes_table

theta	Prior	Likelihood	Product	Posterior
0.0	0.0909091	0.0000000	0.0000000	0.0000000
0.1	0.0909091	0.0000000	0.0000000	0.0000000
0.2	0.0909091	0.0000000	0.0000000	0.0000000
0.3	0.0909091	0.0105527	0.0009593	0.9999971
0.4	0.0909091	0.0000000	0.0000000	0.0000029
0.5	0.0909091	0.0000000	0.0000000	0.0000000
0.6	0.0909091	0.0000000	0.0000000	0.0000000
0.7	0.0909091	0.0000000	0.0000000	0.0000000
0.8	0.0909091	0.0000000	0.0000000	0.0000000
0.9	0.0909091	0.0000000	0.0000000	0.0000000
1.0	0.0909091	0.0000000	0.0000000	0.0000000

NNB Insert discussion of post dist

No I do not think that the prior is reasonable because I don’t think it is reasonable to hold the prior belief that all θ values are equally. A person should hold the prior information that θ close to 0 or 1 is far less likely than values closer to 0.5 based on common sense and political accumen by leading a campaign.

I will choose the following discrete prior instead based on that I strongly believe that θ ∈ [0.3, 0.45] is much more likely than θ is not an element of [0.3,0.45].

p(θ) = {0.001,0.05,0.1,0.25,0.25,0.13,0.1,0.066,0.05,0.002,0.001}

A valid distribution has the following properties: 1. 0 <=p(θ)<= 1 2. Summation of p(θ) for all θ =1

bayes_table_b=data.frame(
  theta_b=seq(0,1,by=0.1),
  Prior=c(0.001,0.05,0.1,0.25,0.25,0.13,0.1,0.066,0.05,0.002,0.001)
)#creating data frame with new prior distribution

max(bayes_table_b$Prior)#max value of p(θ)<=1

## [1] 0.25

min(bayes_table_b$Prior)#min value of p(θ)>=0

## [1] 0.001

sum(bayes_table_b$Prior) # sum of p(θ) for all θ=1

## [1] 1

Therefore my prior is a valid distribution.

I will evaluate the new posterior distribution by using the same method as in part A.

bayes_table_b$Likelihood=dbinom(320,1000,prob = bayes_table_b$theta_b) #finding p(y|θ) 
bayes_table_b=bayesian_crank(bayes_table_b)#finding p(θ|y)
kable(bayes_table_b) #plotting table of posterior

theta_b	Prior	Likelihood	Product	Posterior
0.0	0.001	0.0000000	0.0000000	0.0000000
0.1	0.050	0.0000000	0.0000000	0.0000000
0.2	0.100	0.0000000	0.0000000	0.0000000
0.3	0.250	0.0105527	0.0026382	0.9999971
0.4	0.250	0.0000000	0.0000000	0.0000029
0.5	0.130	0.0000000	0.0000000	0.0000000
0.6	0.100	0.0000000	0.0000000	0.0000000
0.7	0.066	0.0000000	0.0000000	0.0000000
0.8	0.050	0.0000000	0.0000000	0.0000000
0.9	0.002	0.0000000	0.0000000	0.0000000
1.0	0.001	0.0000000	0.0000000	0.0000000

prior_post_plot(bayes_table_b)#plotting prior and post

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## ℹ The deprecated feature was likely used in the ProbBayes package.
##   Please report the issue to the authors.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

NNB Discuss findings and comment on the utility of discrete prior(how cant be 0.45)

Explain why beta is a reasonable choice of prior

I am going to use the beta.select function to choose shape parameters a and b to reflect my belief that that θ ∈ [0.3, 0.45] is more likely than θ is not an element of [0.3, 0.45]. I’m going to specify that

Bayes Assignment 1 Draft

2025-10-21