assume a generative model with bernoulli data x and a uniform prior xbernoulli uniform01 also as

Name: assume a generative model with bernoulli data x and a uniform prior xbernoulli uniform01 also assume you observe data representing two successes and one failure using greta update your prior 01379
Uploaded: 2022-11-18T11:15:18-08:00
Duration: 2 min 17 s
Channel: Sri K
Description: assume a generative model with bernoulli data x and a uniform prior xbernoulli uniform01 also assume you observe data representing two successes and one failure using greta update your prior 01379

First, we need to define our observed data and our prior. In this case, we have observed two suc

Question

Assume a generative model with Bernoulli data (X) and a uniform prior (θ): X ~ Bernoulli(θ) Θ ~ Uniform(0,1) Also, assume you observe data representing two successes and one failure. Using greta, update your prior for θ and generate a representative sample of 4,000 draws from the posterior distribution (note: 4,000 is the default number of samples when using greta's mcmc function to get the posterior). As mentioned in the course content, assign the output of the mcmc function to an object named "draws". Use the following three methods to graphically display your posterior distribution for theta: Method 1) ggplot2: library(tidybayes) ## install.packages("tidybayes", dependencies = TRUE) drawsDF = tidy_draws(draws) ## tidybayes function to get dataframe of draws ggplot(drawsDF, aes(x = theta)) + geom_density() ## generate plot Method 2) Use bayesplot function: library(bayesplot) ## this gets installed with greta automatically - if not, use install.packages() mcmc_areas(draws) Method 3) Use bayesplot function: library(bayesplot) ## this gets installed with greta automatically - if not, use install.packages() mcmc_intervals(draws) While it might not be obvious, the last two methods also generate ggplots that you can modify with standard ggplot2 functionality. Modify each of the three plots with a title: + ggtitle("FirstName Lastname - Posterior") Show your plots generated in each method. EXAMPLES FROM CHAPTER MENTIONED ABOVE: # You observe two successes and one failure # Representative sample of 4,000 draws # What is your updated probability that theta is less than or equal to 50% (i.e. P(theta <= 50%))? # Step 1 - Tell greta about your observed random variables (i.e. data) using as_data() library(greta) ## Assume the first two stores are a success and the third store is not (i.e. make up some data) storeData = c(1, 1, 0) ## Tell greta about your observed data X (recall that X represents sales increase data) x = as_data(storeData) # Step 2 - Tell greta about your prior for unobserved random variables ## Tell greta about what values of theta seem plausible prior to observing the data theta = uniform(0, 1) # Step 3 - Tell greta how observed and unobserved RV's are related ## Use the distribution() function to say how the distribution of an observed random variable might depend on an unobserved random variable. This represents the likelihood in Bayesian inference. distribution(x) = bernoulli(theta) # Step 4 - Define an R object that stores a complete greta model ## Use the model() function to tell greta which random variables you want to analyze in the posterior distribution fit = model(theta) ## By convention, fit is often used to store model definitions # Step 5 - Verify the greta model looks okay ## Use the plot() function to see how greta organized your model plot(fit) # Step 6. Create a representative sample of the joint posterior distribution ## This function might take some time ## It runs for about 20 seconds on my machine ## It stores the posterior distribution in an object called "draws" draws = mcmc(fit) library(tidyverse) # Extract samples as a data frame drawsDF = draws %>% as.matrix() %>% as_tibble() drawsDF ## Make a histogram with bins spaced every 0.01 library(tidyverse) ggplot(drawsDF, aes(x = theta)) + geom_histogram(breaks = seq(0, 1, 0.01)) library(tidyverse) # Get prior representative sample numObs = 4000 ## Number of realizations to generate thetaVec = runif(numObs, min = 0, max = 1) # Vector of theta thetaDF = tibble(theta = thetaVec) ## Dataframe for plotting # Create prior dataframe with sampleType column priorDF = thetaDF %>% mutate(sampleType = "prior") # Create posterior dataframe with sampleType column postDF = drawsDF %>% dplyr::select(theta) %>% mutate(sampleType = "posterior") # Create a data frame of draws with a label for prior/posterior priorPostDF = bind_rows(priorDF, postDF) # View some sample rows priorPostDF %>% sample_n(10) # Use geom_density to create a smoothed histogram ggplot(priorPostDF, aes(x = theta)) + geom_density(aes(fill = sampleType), alpha = 0.3) # Percentage of draws where theta is less than or equal to 50% drawsDF %>% mutate(under50 = ifelse(theta > 0.65, 1, 0)) %>% summarize(avgUnder50 = mean(under50))

          Assume a generative model with Bernoulli data (X) and a uniform prior (θ):
X ~ Bernoulli(θ)
Θ ~ Uniform(0,1)
Also, assume you observe data representing two successes and one failure.
Using greta, update your prior for θ and generate a representative sample of 4,000 draws from the posterior distribution (note: 4,000 is the default number of samples when using greta's mcmc function to get the posterior). As mentioned in the course content, assign the output of the mcmc function to an object named "draws".
Use the following three methods to graphically display your posterior distribution for theta:
Method 1) ggplot2:
library(tidybayes)  ## install.packages("tidybayes", dependencies = TRUE)
drawsDF = tidy_draws(draws)  ## tidybayes function to get dataframe of draws
ggplot(drawsDF, aes(x = theta)) + geom_density() ## generate plot
Method 2) Use bayesplot function:
library(bayesplot)  ## this gets installed with greta automatically - if not, use install.packages()
mcmc_areas(draws)
Method 3) Use bayesplot function:
library(bayesplot)  ## this gets installed with greta automatically - if not, use install.packages()
mcmc_intervals(draws)
While it might not be obvious, the last two methods also generate ggplots that you can modify with standard ggplot2 functionality. Modify each of the three plots with a title:
+ ggtitle("FirstName Lastname - Posterior")
Show your plots generated in each method.

EXAMPLES FROM CHAPTER MENTIONED ABOVE:
# You observe two successes and one failure
# Representative sample of 4,000 draws
# What is your updated probability that theta is less than or equal to 50% (i.e. P(theta <= 50%))?
# Step 1 - Tell greta about your observed random variables (i.e. data) using as_data()
library(greta)
## Assume the first two stores are a success and the third store is not (i.e. make up some data)
storeData = c(1, 1, 0)
## Tell greta about your observed data X (recall that X represents sales increase data)
x = as_data(storeData)
# Step 2 - Tell greta about your prior for unobserved random variables
## Tell greta about what values of theta seem plausible prior to observing the data
theta = uniform(0, 1)
# Step 3 - Tell greta how observed and unobserved RV's are related
## Use the distribution() function to say how the distribution of an observed random variable might depend on an unobserved random variable. This represents the likelihood in Bayesian inference.
distribution(x) = bernoulli(theta)
# Step 4 - Define an R object that stores a complete greta model
## Use the model() function to tell greta which random variables you want to analyze in the posterior distribution
fit = model(theta) ## By convention, fit is often used to store model definitions
# Step 5 - Verify the greta model looks okay
## Use the plot() function to see how greta organized your model
plot(fit)
# Step 6. Create a representative sample of the joint posterior distribution
## This function might take some time
## It runs for about 20 seconds on my machine
## It stores the posterior distribution in an object called "draws"
draws = mcmc(fit)
library(tidyverse)
# Extract samples as a data frame
drawsDF = draws %>% as.matrix() %>% as_tibble()
drawsDF
## Make a histogram with bins spaced every 0.01
library(tidyverse)
ggplot(drawsDF, aes(x = theta)) + geom_histogram(breaks = seq(0, 1, 0.01))
library(tidyverse)
# Get prior representative sample
numObs = 4000 ## Number of realizations to generate
thetaVec = runif(numObs, min = 0, max = 1) # Vector of theta
thetaDF = tibble(theta = thetaVec) ## Dataframe for plotting
# Create prior dataframe with sampleType column
priorDF = thetaDF %>% mutate(sampleType = "prior")
# Create posterior dataframe with sampleType column
postDF = drawsDF %>% dplyr::select(theta) %>% mutate(sampleType = "posterior")
# Create a data frame of draws with a label for prior/posterior
priorPostDF = bind_rows(priorDF, postDF)
# View some sample rows
priorPostDF %>% sample_n(10)
# Use geom_density to create a smoothed histogram
ggplot(priorPostDF, aes(x = theta)) + geom_density(aes(fill = sampleType), alpha = 0.3)
# Percentage of draws where theta is less than or equal to 50%
drawsDF %>% mutate(under50 = ifelse(theta > 0.65, 1, 0)) %>% summarize(avgUnder50 = mean(under50))

Added by Juan Antonio B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

11/18/2022

Step 1

In this case, we have observed two successes and one failure, and our prior is a uniform distribution between 0 and 1. ```R library(greta) # Define observed data storeData = c(1, 1, 0) x = as_data(storeData) # Define prior theta = uniform(0, 1) ``` Show more…

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Assume a generative model with Bernoulli data (X) and a uniform prior (θ): X ~ Bernoulli(θ) Θ ~ Uniform(0,1) Also, assume you observe data representing two successes and one failure. Using greta, update your prior for θ and generate a representative sample of 4,000 draws from the posterior distribution (note: 4,000 is the default number of samples when using greta's mcmc function to get the posterior). As mentioned in the course content, assign the output of the mcmc function to an object named "draws". Use the following three methods to graphically display your posterior distribution for theta: Method 1) ggplot2: library(tidybayes) ## install.packages("tidybayes", dependencies = TRUE) drawsDF = tidy_draws(draws) ## tidybayes function to get dataframe of draws ggplot(drawsDF, aes(x = theta)) + geom_density() ## generate plot Method 2) Use bayesplot function: library(bayesplot) ## this gets installed with greta automatically - if not, use install.packages() mcmc_areas(draws) Method 3) Use bayesplot function: library(bayesplot) ## this gets installed with greta automatically - if not, use install.packages() mcmc_intervals(draws) While it might not be obvious, the last two methods also generate ggplots that you can modify with standard ggplot2 functionality. Modify each of the three plots with a title: + ggtitle("FirstName Lastname - Posterior") Show your plots generated in each method. EXAMPLES FROM CHAPTER MENTIONED ABOVE: # You observe two successes and one failure # Representative sample of 4,000 draws # What is your updated probability that theta is less than or equal to 50% (i.e. P(theta <= 50%))? # Step 1 - Tell greta about your observed random variables (i.e. data) using as_data() library(greta) ## Assume the first two stores are a success and the third store is not (i.e. make up some data) storeData = c(1, 1, 0) ## Tell greta about your observed data X (recall that X represents sales increase data) x = as_data(storeData) # Step 2 - Tell greta about your prior for unobserved random variables ## Tell greta about what values of theta seem plausible prior to observing the data theta = uniform(0, 1) # Step 3 - Tell greta how observed and unobserved RV's are related ## Use the distribution() function to say how the distribution of an observed random variable might depend on an unobserved random variable. This represents the likelihood in Bayesian inference. distribution(x) = bernoulli(theta) # Step 4 - Define an R object that stores a complete greta model ## Use the model() function to tell greta which random variables you want to analyze in the posterior distribution fit = model(theta) ## By convention, fit is often used to store model definitions # Step 5 - Verify the greta model looks okay ## Use the plot() function to see how greta organized your model plot(fit) # Step 6. Create a representative sample of the joint posterior distribution ## This function might take some time ## It runs for about 20 seconds on my machine ## It stores the posterior distribution in an object called "draws" draws = mcmc(fit) library(tidyverse) # Extract samples as a data frame drawsDF = draws %>% as.matrix() %>% as_tibble() drawsDF ## Make a histogram with bins spaced every 0.01 library(tidyverse) ggplot(drawsDF, aes(x = theta)) + geom_histogram(breaks = seq(0, 1, 0.01)) library(tidyverse) # Get prior representative sample numObs = 4000 ## Number of realizations to generate thetaVec = runif(numObs, min = 0, max = 1) # Vector of theta thetaDF = tibble(theta = thetaVec) ## Dataframe for plotting # Create prior dataframe with sampleType column priorDF = thetaDF %>% mutate(sampleType = "prior") # Create posterior dataframe with sampleType column postDF = drawsDF %>% dplyr::select(theta) %>% mutate(sampleType = "posterior") # Create a data frame of draws with a label for prior/posterior priorPostDF = bind_rows(priorDF, postDF) # View some sample rows priorPostDF %>% sample_n(10) # Use geom_density to create a smoothed histogram ggplot(priorPostDF, aes(x = theta)) + geom_density(aes(fill = sampleType), alpha = 0.3) # Percentage of draws where theta is less than or equal to 50% drawsDF %>% mutate(under50 = ifelse(theta > 0.65, 1, 0)) %>% summarize(avgUnder50 = mean(under50))