A Bayesian model of water temperature

Fundamentals of Bayesian Data Analysis in R

Rasmus Bååth

Data Scientist

Let's define the model

Let's fit the model

n_ads_shown <- 100
n_visitors <- 13
proportion_clicks <- seq(0, 1, by = 0.01)
pars <- expand.grid(proportion_clicks = proportion_clicks)
pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)

proportion_clicks <- seq(0, 1, by = 0.01)
pars <- expand.grid(proportion_clicks = proportion_clicks)
pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <-
sigma <- 
pars <- expand.grid(proportion_clicks = proportion_clicks)
pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(proportion_clicks = proportion_clicks)
pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

The parameter space

plot(pars, pch=19)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5)

pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5)
pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10)
pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) 
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5)
pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10)
pars$prior <- pars$mu_prior * pars$sigma_prior
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5)
pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10)
pars$prior <- pars$mu_prior * pars$sigma_prior
for(i in 1:nrow(pars)) {

pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5)
pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10)
pars$prior <- pars$mu_prior * pars$sigma_prior
for(i in 1:nrow(pars)) {
    likelihoods <- dnorm(temp, pars$mu[i], pars$sigma[i])
pars$likelihood <- dbinom(n_visitors, 
    size = n_ads_shown, prob = pars$proportion_clicks)
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Let's fit the model

temp <- c(19, 23, 20, 17, 23)
mu <- seq(8, 30, by = 0.5)
sigma <- seq(0.1, 10, by = 0.3)
pars <- expand.grid(mu = mu, sigma = sigma)
pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5)
pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10)
pars$prior <- pars$mu_prior * pars$sigma_prior
for(i in 1:nrow(pars)) {
    likelihoods <- dnorm(temp, pars$mu[i], pars$sigma[i])
    pars$likelihood[i] <- prod(likelihoods)
}
pars$probability <- pars$likelihood * pars$prior
pars$probability <- pars$probability / sum(pars$probability)

Replicate this analysis using zombie data!

Fundamentals of Bayesian Data Analysis in R