Simple Bayesian

Jan 20, 2019 #bayesian

library(tidyverse)
library(ggthemes)
old <- theme_set(theme_tufte() + theme(text = element_text(family = "Menlo")))

# parameters: 
# number of ads shown
# number of visitors & proportional clicks
n_ads_shown <- 100
n_visitors  <- seq(0, 100, 1)
prop_clicks <- seq(0, 1, 0.01)

params <-
  expand.grid(prop_clicks = prop_clicks, n_visitors = n_visitors) %>%
  as_tibble()

# setup joint probability distribution
jpdf <- params %>% 
  mutate(
    # assume prior uniform from 0 to 20%
    prior      = dunif(prop_clicks, min = 0, max = .2), 
    # generative model goes here
    likelihood = dbinom(n_visitors, size = n_ads_shown, prob = prop_clicks), 
    # on Bayes theorem
    probs      = prior * likelihood,
    probs      = probs / sum(probs)
  )
jpdf

## # A tibble: 10,201 x 5
##    prop_clicks n_visitors prior likelihood      probs
##          <dbl>      <dbl> <dbl>      <dbl>      <dbl>
##  1        0             0     5  1         0.0476    
##  2        0.01          0     5  0.366     0.0174    
##  3        0.02          0     5  0.133     0.00632   
##  4        0.03          0     5  0.0476    0.00226   
##  5        0.04          0     5  0.0169    0.000803  
##  6        0.05          0     5  0.00592   0.000282  
##  7        0.06          0     5  0.00205   0.0000979 
##  8        0.07          0     5  0.000705  0.0000336 
##  9        0.08          0     5  0.000239  0.0000114 
## 10        0.09          0     5  0.0000802 0.00000382
## # … with 10,191 more rows

# update prior based on posterior
# conditioned on x visitors
jpdf %>% 
    filter(n_visitors == 6) %>% 
    ggplot(aes(prop_clicks, probs)) + 
    geom_bar(stat = "identity", width = 0.005) + 
    labs(x = "Proportional Clicks", y = "Probability")