Category: statistics

Simulating and visualizing the Monty Hall problem in Python & R

Simulating and visualizing the Monty Hall problem in Python & R

I recently visited a data science meetup where one of the speakers — Harm Bodewes — spoke about playing out the Monty Hall problem with his kids.

The Monty Hall problem is probability puzzle. Based on the American television game show Let’s Make a Deal and its host, named Monty Hall:

You’re given the choice of three doors.

Behind one door sits a prize: a shiny sports car.

Behind the others doors, something shitty, like goats.

You pick a door — say, door 1.

Now, the host, who knows what’s behind the doors, opens one of the other doors — say, door 2 — which reveals a goat.

The host then asks you:
Do you want to stay with door 1,
or
would you like to switch to door 3?

The probability puzzle here is:

Is switching doors the smart thing to do?

Back to my meetup.

Harm — the presenter — had ran the Monty Hall experiment with his kids.

Twenty-five times, he had hidden candy under one of three plastic cups. His kids could then pick a cup, he’d remove one of the non-candy cups they had not picked, and then he’d proposed them to make the switch.

The results he had tracked, and visualized in a simple Excel graph. And here he was presenting these results to us, his Meetup audience.

People (also statisticans) had been arguing whether it is best to stay or switch doors for years. Yet, here, this random guy ran a play-experiment and provided very visual proof removing any doubts you might have yourself.

You really need to switch doors!

At about the same time, I came across this Github repo by Saghir, who had made some vectorised simulations of the problem in R. I thought it was a fun excercise to simulate and visualize matters in two different data science programming languages — Python & R — and see what I’d run in to.

So I’ll cut to the chase.

As we play more and more games against Monty Hall, it becomes very clear that you really, really, really need to switch doors in order to maximize the probability of winning a car.

Actually, the more games we play, the closer the probability of winning in our sample gets to the actual probability.

Even after 1000 games, the probabilities are still not at their actual values. But, ultimately…

If you stick to your door, you end up with the car in only 33% of the cases.

If you switch to the other door, you end up with the car 66% of the time!

Simulation Code

In both Python and R, I wrote two scripts. You can find the most recent version of the code on my Github. However, I pasted the versions of March 4th 2020 below.

The first script contains a function simulating a single game of Monty Hall. A second script runs this function an X amount of times, and visualizes the outcomes as we play more and more games.

Python

simulate_game.py

import random

def simulate_game(make_switch=False, n_doors=3, seed=None):
    ''' 
    Simulate a game of Monty Hall
    For detailed information: https://en.wikipedia.org/wiki/Monty_Hall_problem
    Basically, there are several closed doors and behind only one of them is a prize.
    The player can choose one door at the start. 
    Next, the game master (Monty Hall) opens all the other doors, but one.
    Now, the player can stick to his/her initial choice or switch to the remaining closed door.
    If the prize is behind the player's final choice he/she wins.

    Keyword arguments:
    make_switch -- a boolean value whether the player switches after its initial choice and Monty Hall opening all other non-prize doors but one (default False)
    n_doors -- an integer value > 2, for the number of doors behind which one prize and (n-1) non-prizes (e.g., goats) are hidden (default 3)
    seed -- a seed to set (default None)
    '''

    # check the arguments
    if type(make_switch) is not bool:
        raise TypeError("`make_switch` must be boolean")
    if type(n_doors) is float:
        n_doors = int(n_doors)
        raise Warning("float value provided for `n_doors`: forced to integer value of", n_doors)
    if type(n_doors) is not int:
        raise TypeError("`n_doors` needs to be a positive integer > 2")
    if n_doors < 2:
        raise ValueError("`n_doors` needs to be a positive integer > 2")

    # if a seed was provided, set it
    if seed is not None:
        random.seed(seed)

    # sample one index for the door to hide the car behind
    prize_index = random.randint(0, n_doors - 1)

    # sample one index for the door initially chosen by the player
    choice_index = random.randint(0, n_doors - 1)

    # we can test for the current result
    current_result = prize_index == choice_index

    # now Monty Hall opens all doors the player did not choose, except for one door
    # next, he asks the player if he/she wants to make a switch
    if (make_switch):
        # if we do, we change to the one remaining door, which inverts our current choice
        # if we had already picked the prize door, the one remaining closed door has a nonprize
        # if we had not already picked the prize door, the one remaining closed door has the prize
        return not current_result
    else:
        # the player sticks with his/her original door,
        # which may or may not be the prize door
        return current_result

visualize_game_results.py

from simulate_game import simulate_game
from random import seed
from numpy import mean, cumsum
from matplotlib import pyplot as plt
import os

# set the seed here
# do not set the `seed` parameter in `simulate_game()`,
# as this will make the function retun `n_games` times the same results
seed(1)

# pick number of games you want to simulate
n_games = 1000

# simulate the games and store the boolean results
results_with_switching = [simulate_game(make_switch=True) for _ in range(n_games)]
results_without_switching = [simulate_game(make_switch=False) for _ in range(n_games)]

# make a equal-length list showing, for each element in the results, the game to which it belongs
games = [i + 1 for i in range(n_games)]

# generate a title based on the results of the simulations
title = f'Switching doors wins you {sum(results_with_switching)} of {n_games} games ({mean(results_with_switching) * 100:.1f}%)' + \
    '\n' + \
    f'as opposed to only {sum(results_without_switching)} games ({mean(results_without_switching) * 100:.1f}%) when not switching'

# set some basic plotting parameters
w = 8
h = 5

# make a line plot of the cumulative wins with and without switching
plt.figure(figsize=(w, h))
plt.plot(games, cumsum(results_with_switching), color='blue', label='switching')
plt.plot(games, cumsum(results_without_switching), color='red', label='no switching')
plt.axis([0, n_games, 0, n_games])
plt.title(title)
plt.legend()
plt.xlabel('Number of games played')
plt.ylabel('Cumulative number of games won')
plt.figtext(0.95, 0.03, 'paulvanderlaken.com', wrap=True, horizontalalignment='right', fontsize=6)

# you can uncomment this to see the results directly,
# but then python will not save the result to your directory
# plt.show()
# plt.close()

# create a directory to store the plots in
# if this directory does not yet exist
try:
    os.makedirs('output')
except OSError:
    None
plt.savefig('output/monty-hall_' + str(n_games) + '_python.png')

Visualizations (matplotlib)

R

simulate-game.R

Note that I wrote a second function, simulate_n_games, which just runs simulate_game an N number of times.

#' Simulate a game of Monty Hall
#' For detailed information: https://en.wikipedia.org/wiki/Monty_Hall_problem
#' Basically, there are several closed doors and behind only one of them is a prize.
#' The player can choose one door at the start. 
#' Next, the game master (Monty Hall) opens all the other doors, but one.
#' Now, the player can stick to his/her initial choice or switch to the remaining closed door.
#' If the prize is behind the player's final choice he/she wins.
#' 
#' @param make_switch A boolean value whether the player switches after its initial choice and Monty Hall opening all other non-prize doors but one. Defaults to `FALSE`
#' @param n_doors An integer value > 2, for the number of doors behind which one prize and (n-1) non-prizes (e.g., goats) are hidden. Defaults to `3L`
#' @param seed A seed to set. Defaults to `NULL`
#'
#' @return A boolean value indicating whether the player won the prize
#'
#' @examples 
#' simulate_game()
#' simulate_game(make_switch = TRUE)
#' simulate_game(make_switch = TRUE, n_doors = 5L, seed = 1)
simulate_game = function(make_switch = FALSE, n_doors = 3L, seed = NULL) {
  
  # check the arguments
  if (!is.logical(make_switch) | is.na(make_switch)) stop("`make_switch` needs to be TRUE or FALSE")
  if (is.double(n_doors)) {
    n_doors = as.integer(n_doors)
    warning(paste("double value provided for `n_doors`: forced to integer value of", n_doors))
  }
  if (!is.integer(n_doors) | n_doors < 2) stop("`n_doors` needs to be a positive integer > 2")
  
  # if a seed was provided, set it
  if (!is.null(seed)) set.seed(seed)
  
  # create a integer vector for the door indices
  doors = seq_len(n_doors)
  
  # create a boolean vector showing which doors are opened
  # all doors are closed at the start of the game
  isClosed = rep(TRUE, length = n_doors)
  
  # sample one index for the door to hide the car behind
  prize_index = sample(doors, size = 1)
  
  # sample one index for the door initially chosen by the player
  # this can be the same door as the prize door
  choice_index = sample(doors, size = 1)
  
  # now Monty Hall opens all doors the player did not choose
  # except for one door
  # if we have already picked the prize door, the one remaining closed door has a nonprize
  # if we have not picked the prize door, the one remaining closed door has the prize
  if (prize_index == choice_index) {
    # if we have the prize, Monty Hall can open all but two doors:
    #   ours, which we remove from the options to sample from and open
    #   and one goat-conceiling door, which we do not open
    isClosed[sample(doors[-prize_index], size = n_doors - 2)] = FALSE
  } else {
    # else, Monty Hall can also open all but two doors:
    #   ours
    #   and the prize-conceiling door
    isClosed[-c(prize_index, choice_index)] = FALSE
  }
  
  # now Monty Hall asks us whether we want to make a switch
  if (make_switch) {
    # if we decide to make a switch, we can pick the closed door that is not our door
    choice_index = doors[isClosed][doors[isClosed] != choice_index]
  }
  
  # we return a boolean value showing whether the player choice is the prize door
  return(choice_index == prize_index)
}


#' Simulate N games of Monty Hall
#' Calls the `simulate_game()` function `n` times and returns a boolean vector representing the games won
#' 
#' @param n An integer value for the number of times to call the `simulate_game()` function
#' @param seed A seed to set in the outer loop. Defaults to `NULL`
#' @param ... Any parameters to be passed to the `simulate_game()` function. 
#' No seed can be passed to the simulate_game function as that would result in `n` times the same result 
#'
#' @return A boolean vector indicating for each of the games whether the player won the prize
#'
#' @examples 
#' simulate_n_games(n = 100)
#' simulate_n_games(n = 500, make_switch = TRUE)
#' simulate_n_games(n = 1000, seed = 123, make_switch = TRUE, n_doors = 5L)
simulate_n_games = function(n, seed = NULL, make_switch = FALSE, ...) {
  # round the number of iterations to an integer value
  if (is.double(n)) {
    n = as.integer(n)
  }
  if (!is.integer(n) | n < 1) stop("`n_games` needs to be a positive integer > 1")
  # if a seed was provided, set it
  if (!is.null(seed)) set.seed(seed)
  return(vapply(rep(make_switch, n), simulate_game, logical(1), ...))
}

visualize-game-results.R

Note that we source in the simulate-game.R file to get access to the simulate_game and simulate_n_games functions.

Also note that I make a second plot here, to show the probabilities of winning converging to their real-world probability as we play more and more games.

source('R/simulate-game.R')

# install.packages('ggplot2')
library(ggplot2)

# set the seed here
# do not set the `seed` parameter in `simulate_game()`,
# as this will make the function return `n_games` times the same results
seed = 1

# pick number of games you want to simulate
n_games = 1000

# simulate the games and store the boolean results
results_without_switching = simulate_n_games(n = n_games, seed = seed, make_switch = FALSE)
results_with_switching = simulate_n_games(n = n_games, seed = seed, make_switch = TRUE)

# store the cumulative wins in a dataframe
results = data.frame(
  game = seq_len(n_games),
  cumulative_wins_without_switching = cumsum(results_without_switching),
  cumulative_wins_with_switching = cumsum(results_with_switching)
)

# function that turns values into nice percentages
format_percentage = function(values, digits = 1) {
  return(paste0(formatC(values * 100, digits = digits, format = 'f'), '%'))
}

# generate a title based on the results of the simulations
title = paste(
  paste0('Switching doors wins you ', sum(results_with_switching), ' of ', n_games, ' games (', format_percentage(mean(results_with_switching)), ')'),
  paste0('as opposed to only ', sum(results_without_switching), ' games (', format_percentage(mean(results_without_switching)), ') when not switching)'),
  sep = '\n'
)

# set some basic plotting parameters
linesize = 1 # size of the plotted lines
x_breaks = y_breaks = seq(from = 0, to = n_games, length.out = 10 + 1) # breaks of the axes
y_limits = c(0, n_games) # limits of the y axis - makes y limits match x limits
w = 8 # width for saving plot
h = 5 # height for saving plot
palette = setNames(c('blue', 'red'), nm = c('switching', 'without switching')) # make a named color scheme

# make a line plot of the cumulative wins with and without switching
ggplot(data = results) +
  geom_line(aes(x = game, y = cumulative_wins_with_switching, col = names(palette[1])), size = linesize) +
  geom_line(aes(x = game, y = cumulative_wins_without_switching, col = names(palette[2])), size = linesize) +
  scale_x_continuous(breaks = x_breaks) +
  scale_y_continuous(breaks = y_breaks, limits = y_limits) +
  scale_color_manual(values = palette) +
  theme_minimal() +
  theme(legend.position = c(1, 1), legend.justification = c(1, 1), legend.background = element_rect(fill = 'white', color = 'transparent')) +
  labs(x = 'Number of games played') +
  labs(y = 'Cumulative number of games won') +
  labs(col = NULL) +
  labs(caption = 'paulvanderlaken.com') +
  labs(title = title)

# save the plot in the output folder
# create the output folder if it does not exist yet
if (!file.exists('output')) dir.create('output', showWarnings = FALSE)
ggsave(paste0('output/monty-hall_', n_games, '_r.png'), width = w, height = h)


# make a line plot of the rolling % win chance with and without switching
ggplot(data = results) +
  geom_line(aes(x = game, y = cumulative_wins_with_switching / game, col = names(palette[1])), size = linesize) +
  geom_line(aes(x = game, y = cumulative_wins_without_switching / game, col = names(palette[2])), size = linesize) +
  scale_x_continuous(breaks = x_breaks) +
  scale_y_continuous(labels = function(x) format_percentage(x, digits = 0)) +
  scale_color_manual(values = palette) +
  theme_minimal() +
  theme(legend.position = c(1, 1), legend.justification = c(1, 1), legend.background = element_rect(fill = 'white', color = 'transparent')) +
  labs(x = 'Number of games played') +
  labs(y = '% of games won') +
  labs(col = NULL) +
  labs(caption = 'paulvanderlaken.com') +
  labs(title = title)


# save the plot in the output folder
# create the output folder if it does not exist yet
if (!file.exists('output')) dir.create('output', showWarnings = FALSE)
ggsave(paste0('output/monty-hall_perc_', n_games, '_r.png'), width = w, height = h)

Visualizations (ggplot2)

I specifically picked a seed (the second one I tried) in which not switching looked like it was better during the first few games played.

In R, I made an additional plot that shows the probabilities converging.

As we play more and more games, our results move to the actual probabilities of winning:

After the first four games, you could have erroneously concluded that not switching would result in better chances of you winning a sports car. However, in the long run, that is definitely not true.

I was actually suprised to see that these lines look to be mirroring each other. But actually, that’s quite logical maybe… We already had the car with our initial door guess in those games. If we would have sticked to that initial choice of a door, we would have won, whereas all the cases where we switched, we lost.

Keep me posted!

I hope you enjoyed these simulations and visualizations, and am curious to see what you come up with yourself!

For instance, you could increase the number of doors in the game, or the number of goat-doors Monty Hall opens. When does it become a disadvantage to switch?

Cover image via Medium

Solutions to working with small sample sizes

Solutions to working with small sample sizes

Both in science and business, we often experience difficulties collecting enough data to test our hypotheses, either because target groups are small or hard to access, or because data collection entails prohibitive costs.

Such obstacles may result in data sets that are too small for the complexity of the statistical model needed to answer the questions we’re really interested in.

Several scholars teamed up and wrote this open access book: Small Sample Size Solutions.

This unique book provides guidelines and tools for implementing solutions to issues that arise in small sample studies. Each chapter illustrates statistical methods that allow researchers and analysts to apply the optimal statistical model for their research question when the sample is too small.

This book will enable anyone working with data to test their hypotheses even when the statistical model required for answering their questions are too complex for the sample sizes they can collect. The covered statistical models range from the estimation of a population mean to models with latent variables and nested observations, and solutions include both classical and Bayesian methods. All proposed solutions are described in steps researchers can implement with their own data and are accompanied with annotated syntax in R.

You can access the book for free here!

Probability Distributions mapped and explained by their relationships

Probability Distributions mapped and explained by their relationships

Sean Owen created this handy cheat sheet that shows the most common probability distributions mapped by their underlying relationships.

Probability distributions are fundamental to statistics, just like data structures are to computer science. They’re the place to start studying if you mean to talk like a data scientist. 

Sean Owen (via)

Owen argues that the probability distributions relate to each other in intuitive and interesting ways that makes it easier for you to recall them. For instance, several follow naturally from the Bernoulli distribution. Having this map by hand should thus help you really understand what these distributions imply.

On top of that, it’s just a nice geeky network poster!

Sean’s map of the relationships between probability distributions (via)

Now, Sean didn’t just make a fancy map. In the original blog he also explains each of the distributions and how it relates to the others. Having this knowledge is vital to being a good data scientist / analyst.

You can sometimes get away with simple analysis using R or scikit-learn without quite understanding distributions, just like you can manage a Java program without understanding hash functions. But it would soon end in tears, bugs, bogus results, or worse: sighs and eye-rolling from stats majors.

Sean Owen (via)

For instance, here’s Sean explaining the Binomial distribution:

The binomial distribution may be thought of as the sum of outcomes of things that follow a Bernoulli distribution. Toss a fair coin 20 times; how many times does it come up heads? This count is an outcome that follows the binomial distribution. Its parameters are n, the number of trials, and p, the probability of a “success” (here: heads, or 1). Each flip is a Bernoulli-distributed outcome, or trial. Reach for the binomial distribution when counting the number of successes in things that act like a coin flip, where each flip is independent and has the same probability of success.

Sean Owen (via)

Header image via Alison-Static

Simulating data with Bayesian networks, by Daniel Oehm

Simulating data with Bayesian networks, by Daniel Oehm

Daniel Oehm wrote this interesting blog about how to simulate realistic data using a Bayesian network.

Bayesian networks are a type of probabilistic graphical model that uses Bayesian inference for probability computations. Bayesian networks aim to model conditional dependence, and therefore causation, by representing conditional dependence by edges in a directed graph. Through these relationships, one can efficiently conduct inference on the random variables in the graph through the use of factors.

Devin Soni via Medium

As Bayes nets represent data as a probabilistic graph, it is very easy to use that structure to simulate new data that demonstrate the realistic patterns of the underlying causal system. Daniel’s post shows how to do this with bnlearn.

Daniel’s example Bayes net

New data is simulated from a Bayes net (see above) by first sampling from each of the root nodes, in this case sex. Then followed by the children conditional on their parent(s) (e.g. sport | sex and hg | sex) until data for all nodes has been drawn. The numbers on the nodes below indicate the sequence in which the data is simulated, noting that rcc is the terminal node.

Daniel Oehms in his blog

The original and simulated datasets are compared in a couple of ways 1) observing the distributions of the variables 2) comparing the output from various models and 3) comparing conditional probability queries. The third test is more of a sanity check. If the data is generated from the original Bayes net then a new one fit on the simulated data should be approximately the same. The more rows we generate the closer the parameters will be to the original values.

The original data alongside the generated data in Daniel’s example

As you can see, a Bayesian network allows you to generate data that looks, feels, and behaves a lot like the data on which you based your network on in the first place.

This can be super useful if you want to generate a synthetic / fake / artificial dataset without sharing personal or sensitive data.

Moreover, the underlying Bayesian net can be very useful to compute missing values. In Daniel’s example, he left out some values on purpose (pretending they were missing) and imputed them with the Bayes net. He found that the imputed values for the missing data points were quite close to the original ones:

For two variables, the original values plotted against the imputed replacements.

In the original blog, Daniel goes on to show how to further check the integrity of the simulated data using statistical models and shares all his code so you can try this out yourself. Please do give his website a visit as Daniel has many more interesting statistics blogs!

Learn Julia for Data Science

Learn Julia for Data Science

Most data scientists favor Python as a programming language these days. However, there’s also still a large group of data scientists coming from a statistics, econometrics, or social science and therefore favoring R, the programming language they learned in university. Now there’s a new kid on the block: Julia.

Image result for julia programming"
Via Medium

Advantages & Disadvantages

According to some, you can think of Julia as a mixture of R and Python, but faster. As a programming language for data science, Julia has some major advantages:

  1. Julia is light-weight and efficient and will run on the tiniest of computers
  2. Julia is just-in-time (JIT) compiled, and can approach or match the speed of C
  3. Julia is a functional language at its core
  4. Julia support metaprogramming: Julia programs can generate other Julia programs
  5. Julia has a math-friendly syntax
  6. Julia has refined parallelization compared to other data science languages
  7. Julia can call C, Fortran, Python or R packages

However, others also argue that Julia comes with some disadvantages for data science, like data frame printing, 1-indexing, and its external package management.

Comparing Julia to Python and R

Open Risk Manual published this side-by-side review of the main open source Data Science languages: Julia, Python, R.

You can click the links below to jump directly to the section you’re interested in. Once there, you can compare the packages and functions that allow you to perform Data Science tasks in the three languages.

GeneralDevelopmentAlgorithms & Datascience
History and CommunityDevelopment EnvironmentGeneral Purpose Mathematical Libraries
Devices and Operating SystemsFiles, Databases and Data ManipulationCore Statistics Libraries
Package ManagementWeb, Desktop and Mobile DeploymentEconometrics / Timeseries Libraries
Package DocumentationSemantic Web / Semantic DataMachine Learning Libraries
Language CharacteristicsHigh Performance ComputingGeoSpatial Libraries
Using R, Python and Julia togetherVisualization
Via openriskmanual.org/wiki/Overview_of_the_Julia-Python-R_Universe

Starting with Julia for Data Science

Here’s a very well written Medium article that guides you through installing Julia and starting with some simple Data Science tasks. At least, Julia’s plots look like:

Via Medium
Bayes theorem, and making probability intuitive – by 3Blue1Brown

Bayes theorem, and making probability intuitive – by 3Blue1Brown

This video I’ve been meaning to watch for a while now. It another great visual explanation of a statistics topic by the 3Blue1Brown Youtube channel (which I’ve covered before, multiple times).

This time, it’s all about Bayes theorem, and I just love how Grant Sanderson explains the concept so visually. He argues that rather then memorizing the theorem, we’d rather learn how to draw out the context. Have a look at the video, or read my summary below:

Grant Sanderson explains the concept very visually following an example outlined in Daniel Kahneman’s and Amos Tversky’s book Thinking Fast, Thinking Slow:

Steve is very shy and withdrawn, invariably helpful but with very little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.”

Is Steve more likely to be a librarian or a farmer?

Question from Thinking Fast, Thinking Slow

What was your first guess?

Kahneman and Tversky argue that people take into account Steve’s disposition and therefore lean towards librarians.

However, few people take into account that librarians are quite scarce in our society, which is rich with farmers. For every librarian, there are 20+ farmers. Hence, despite the disposition, Steve is probably more like to be a farmer.

https://www.youtube.com/watch?v=HZGCoVF3YvM&feature=youtu.be
https://www.youtube.com/watch?v=HZGCoVF3YvM&feature=youtu.be
https://www.youtube.com/watch?v=HZGCoVF3YvM&feature=youtu.be

Rather than remembering the upper theorem, Grant argues that it’s often easier to just draw out the rectangle of probabilities below.

Try it out for yourself using another example by Kahneman and Tversky:

https://www.youtube.com/watch?v=HZGCoVF3YvM&feature=youtu.be