Thinking Like a Bayesian

Introduction

• On Monday:
  • Refresher on probability theory
  • What does each distribution do?
    • beta \to outcomes limited to [0, 1]
    • binomial \to binary outcomes
    • gamma \to continuous & positive outcomes; skewed right
    • normal \to continuous outcomes; mound-shaped & symmetric distribution
    • Poisson \to count outcomes; skewed right
    • uniform \to each outcome has equal probability; rectangular distribution
• Today: building up Bayesian analysis concepts

R Packages Needed

• To follow today’s lecture, please load the following packages:
  • tidyverse
  • janitor
  • bayesrules

library(tidyverse)
library(janitor)
library(bayesrules)

Thinking Like a Bayesian

• Bayesian analysis involves updating beliefs based on observed data.

Thinking Like a Bayesian

• Bayesian analysis involves updating beliefs based on observed data.

Thinking Like a Bayesian

• Bayesian analysis involves updating beliefs based on observed data.

Thinking Like a Bayesian

• Bayesian analysis involves updating beliefs based on observed data.

Thinking Like a Bayesian

• This is the natural Bayesian knowledge-building process of:
  • acknowledging your preconceptions (prior distribution),

Thinking Like a Bayesian

• This is the natural Bayesian knowledge-building process of:
  • acknowledging your preconceptions (prior distribution),
  • using data (data distribution) to update your knowledge (posterior distribution),

Thinking Like a Bayesian

• This is the natural Bayesian knowledge-building process of:
  • acknowledging your preconceptions (prior distribution),
  • using data (data distribution) to update your knowledge (posterior distribution), and
  • repeating (posterior distribution \to new prior distribution)

Thinking Like a Bayesian

• Bayesian and frequentist analyses share a common goal: to learn from data about the world around us.
  • Both Bayesian and frequentist analyses use data to fit models, make predictions, and evaluate hypotheses.
  • When working with the same data, they will typically produce a similar set of conclusions.
• Statisticians typically identify as either a “Bayesian” or “frequentist” …
  • We are not going to “take sides.”
  • We will see these as tools in our toolbox.

Thinking Like a Bayesian

• Bayesian probability: the relative plausibility of an event. This considers prior belief.

Thinking Like a Bayesian

• Frequentist probability: the long-run relative frequency of a repeatable event. This does not consider prior belief.

Thinking Like a Bayesian

• The Bayesian framework depends upon prior information, data, and the balance between them.

  • The balance between the prior information and data is determined by the relative strength of each.

• When we have little data, our posterior can rely more on prior knowledge.

• As we collect more data, the prior can lose its influence.

Thinking Like a Bayesian

• We can also use this approach to combine analysis results.

Thinking Like a Bayesian

• We will use an example to work through Bayesian logic.

• The Collins Dictionary named “fake news” the 2017 term of the year.

  • Fake, misleading, and biased news has proliferated along with online news and social media platforms which allow users to post articles with little quality control.

• We want to flag articles as “real” or “fake.”

• We’ll examine a sample of 150 articles which were posted on Facebook and fact checked by five BuzzFeed journalists (Shu et al. 2017).

Thinking Like a Bayesian

• Information about each article is stored in the fake_news dataset in the bayesrules package.

fake_news <- bayesrules::fake_news
print(colnames(fake_news))

 [1] "title"                   "text"                   
 [3] "url"                     "authors"                
 [5] "type"                    "title_words"            
 [7] "text_words"              "title_char"             
 [9] "text_char"               "title_caps"             
[11] "text_caps"               "title_caps_percent"     
[13] "text_caps_percent"       "title_excl"             
[15] "text_excl"               "title_excl_percent"     
[17] "text_excl_percent"       "title_has_excl"         
[19] "anger"                   "anticipation"           
[21] "disgust"                 "fear"                   
[23] "joy"                     "sadness"                
[25] "surprise"                "trust"                  
[27] "negative"                "positive"               
[29] "text_syllables"          "text_syllables_per_word"

Thinking Like a Bayesian

• Suppose that the most recent article posted to a social media platform is titled: The president has a funny secret!
  • Some features of this title probably set off some red flags.
  • For example, the usage of an exclamation point might seem like an odd choice for a real news article.
• In the dataset, what is the split of real and fake articles?

fake_news %>% tabyl(type)

Thinking Like a Bayesian

• In the dataset, what is the split of real and fake articles?

fake_news %>% tabyl(type)

Thinking Like a Bayesian

• What does the data say?

fake_news %>% tabyl(title_has_excl, type) %>%
  adorn_percentages("col") %>%
  adorn_pct_formatting(digits = 2) %>%
  adorn_ns()

Thinking Like a Bayesian

• What does the data say?

fake_news %>% tabyl(title_has_excl, type) %>%
  adorn_percentages("col") %>%
  adorn_pct_formatting(digits = 2) %>%
  adorn_ns()

• 26.67% of fake news titles use ! vs 2.22% of real news titles use !

Thinking Like a Bayesian

• We now have two pieces of contradictory information.
  • Our prior information suggested that incoming articles are most likely real.
  • However, the exclamation point data is more consistent with fake news.

Building a Bayesian Model

• Thinking like Bayesians, we know that balancing both pieces of information is important in developing a posterior understanding of whether the article is fake.

• Our fake news analysis studies two variables:

  • an article’s fake vs real status and
  • its use of exclamation points.

• We can represent the randomness in these variables using probability models.

Building a Bayesian Model

• Let’s now formalize our prior understanding of whether the new article is fake.

• Based on our data, we saw that 40% of articles are fake and 60% are real.

  • Before reading the new article, there’s a 0.4 prior probability that it’s fake.

P\left[B\right] = 0.40 \text{ and } P\left[B^c\right] = 0.60

• Remember that a valid probability model must:

  1. account for all possible events
  2. assign prior probabilities to each event
  3. have probabilities that sum to 1

Building a Bayesian Model

• Revisiting the contingency table for conditional probabilities,

• If an article is fake (B), what is the probability that it uses exclamation points in the title?
  
  
  
  

• If an article is real (B^c), what is the probability that it uses exclamation points in the title?

Building a Bayesian Model

• We have the following conditional probabilities:
  • If an article is fake, then there is about a 27% chance it uses exclamation points in the title.
    • P[A|B]=0.2667
  • If an article is real, then there is only a 2% chance it uses exclamation points in the title.
    • P[A|B^c]=0.0222
• Exclamation point usage is much more likely among fake news than real news.
  • We have evidence that the article is fake.

Building a Bayesian Model

• Note that we know that the incoming article used exclamation points (A), but we do not actually know if the article is fake (B or B^c).

• In this case, we compared P[A|B] and P[A|B^c] to ascertain the relative likelihood of observing A under different scenarios. That is,

  • L\left[B|A\right] = P\left[A|B\right] \to the likelihood of fake given !
  • L\left[B^c|A\right] = P\left[A|B^c\right] \to the likelihood of real given !

Building a Bayesian Model

L\left[B|A\right] = P\left[A|B\right] \text{ and } L\left[B^c|A\right] = P\left[A|B^c\right]

• It is important for us to note that the likelihood function is not a probability function.
  • This is a framework to determine the relative compatibility of our exclamation point data with B and B^c.

Building a Bayesian Model

• The prior evidence suggested the article is most likely real,

P[B] = 0.4 < P[B^c] = 0.6

• The data, however, is more consistent with the article being fake,

L[B|A] = 0.2667 > L[B^c|A] = 0.0222

Building a Bayesian Model

• We can summarize our probabilities in a table,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

• Find P[A \cap B].

Building a Bayesian Model

• We can summarize our probabilities in a table,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

\begin{align*} P[A \cap B] &= P[A|B] \times P[B] \\ &= 0.2667 \times 0.4 \\ &= 0.1067 \end{align*}

Building a Bayesian Model

• Here’s what we know,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

• Find P[A^c \cap B].

Building a Bayesian Model

• Here’s what we know,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

\begin{align*} P[A^c \cap B] &= P[A^c|B] \times P[B] \\ &= \left(1-P[A|B]\right) \times P[B] \\ &= (1-0.2667) \times 0.4 \\ &= 0.2933 \end{align*}

Building a Bayesian Model

• Here’s what we know,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

• Find P[A \cap B^c].

Building a Bayesian Model

• Here’s what we know,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

\begin{align*} P[A \cap B^c] &= P[A|B^c] \times P[B^c] \\ &= 0.0222 \times 0.6 \\ &= 0.0133 \end{align*}

Building a Bayesian Model

• Here’s what we know,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

• Find P[A^c \cap B^c].

Building a Bayesian Model

• Here’s what we know,

• As found earlier, P[A|B] = 0.2667 and P[A|B^c]=0.0222.

\begin{align*} P[A^c \cap B^c] &= P[A^c|B^c] \times P[B^c] \\ &= 0.9778 \times 0.6 \\ &= 0.5867 \end{align*}

Building a Bayesian Model

• Here’s what we know,

• Finally,

\begin{align*} &P[A] = 0.1067 + 0.0133 = 0.12 \\ &P[A^c] = 0.2933 + 0.5867 = 0.88 \end{align*}

Building a Bayesian Model

• Using rules of probability, we have completed the table.

Building a Bayesian Model

• With more information, we can finally find the probability that the latest article is fake.

• We will use the posterior probability, P[B|A], which is found using Bayes’ Rule.

• Bayes’ Rule: For events A and B,

P[B|A] = \frac{P[A \cap B]}{P[A]} = \frac{P[B] \times L[B|A]}{P[A]}

• But really, we can think about it like this,

\text{posterior} = \frac{\text{prior} \times \text{likelihood}}{\text{normalizing constant}}

Building a Bayesian Model

• Let’s now apply Bayes’ Rule to our fake news example.

• Find P[B|A].

Building a Bayesian Model

• Applying Bayes’ Rule to our fake news example,

P[B|A] = \frac{P[A \cap B]}{P[A]} = \frac{0.1067}{0.12} = 0.889

• There is an 88.9% posterior probability that the article is fake given that it uses an exclamation point in the title.

Building a Bayesian Model

• Comparing our prior and posterior probabilities,
  • Prior: P[B] = 0.40, i.e., before seeing the title, 40% chance the article is fake.
  • Posterior: P[B|A] = 0.889, i.e., after seeing the exclamation point in the title, 88.9% chance the article is fake.
• The exclamation point adds strong evidence to the article being fake.
  • Our posterior is much higher than our prior.

Example 2: Building a Model

• Suppose you’re watching the pilot of a new TV show. Before learning anything about the main character, we can use information from the U.S. Census: the Midwest (M), Northeast (N), South (S), or West (W).

• What is the probability that the main character is from the South?

Example 2: Building a Model

• Suppose you’re watching the pilot of a new TV show. Before learning anything about the main character, we can use information from the U.S. Census: the Midwest (M), Northeast (N), South (S), or West (W).

• During the opening scene, the main character asks someone if they would like a “Pop”.

• What is the probability that the main character is from the South?

Example 2: Building a Model

• Suppose you’re watching the pilot of a new TV show. Before learning anything about the main character, we can use information from the U.S. Census: the Midwest (M), Northeast (N), South (S), or West (W).

• During the opening scene, the main character asks someone if they would like a “Coke”.

• What is the probability that the main character is from the South?

Example 2: Building a Model

• Let’s use the pop_vs_soda dataset to explore the relationship between region (region) and whether people say “pop” or “soda” when referring to carbonated beverages (pop).

data(pop_vs_soda)
pop_vs_soda %>% 
  tabyl(pop, region) %>% 
  adorn_percentages("col") %>%
  round(4)

pop_vs_soda %>% tabyl(pop)

Example 2: Building a Model

• Let’s use the pop_vs_soda dataset to explore the relationship between region (region) and whether people say “pop” or “soda” when referring to carbonated beverages (pop).

• Let A be the event that a person uses the word “pop”. Rewrite the table as regional likelihoods.
  • L[M|A] =
  • L[N|A] =
  • L[S|A] =
  • L[W|A] =

Example 2: Building a Model

• Let A be the event that a person uses the word “pop”. Rewrite the table as regional likelihoods.
  • L[M|A] = P[A|M] = 0.6447 – 64.5% of Midwesterners say “pop”
  • L[N|A] = P[A|N] = 0.2734 – 27.3% of Northeasterners say “pop”
  • L[S|A] = P[A|S] = 0.0792 – 7.9% of Southerners say “pop”
  • L[W|A] = P[A|W] = 0.2943 – 29.4% of Westerners say “pop”
• If the main character says “pop”, we can use the likelihoods to update our prior beliefs about where they are from.

Example 2: Building a Model

• Let’s now formally find the posterior probabilities for the main character being from any of the areas. Complete the following table:

Example 2: Building a Model

• Posterior probability for the midwest:

Example 2: Building a Model

• Posterior probability for the northeast:

Example 2: Building a Model

• Posterior probability for the south:

Example 2: Building a Model

• Posterior probability for the west:

Example 2: Building a Model

• Thus, our table is

• After we hear the main character say “pop,” we now think it’s most likely that they live in the Midwest.
  • Even though the South has the highest proportion of people in the survey, it is least likely that the main character lives in the South.

Example 3: Building a Model

• In 1996, Gary Kasparov played a six-game chess match against the IBM supercomputer Deep Blue.

  • Of the six games, Kasparov won three, drew two, and lost one.
  • Thus, Kasparov won the overall match.

• Kasparov and Deep Blue were to meet again for a six-game match in 1997.

• Let \pi denote Kasparov’s chances of winning any particular game in the re-match.

  • Thus, \pi is a measure of his overall skill relative to Deep Blue.
  • Given the complexity of chess, machines, and humans, \pi is unknown and can vary over time.
    • i.e., \pi is a random variable.

Example 3: Building a Model

• Our first step is to start with a prior model. This model
  • Identifies what values \pi can take,
  • assigns a prior weight or probability to each, and
  • these probabilities sum to 1.
• Based on what we were told, the prior model for \pi in our example,

Example 3: Building a Model

• Based on what we were told, the prior model for \pi in our example,

• Note that this is an incredibly simple model.
  • The win probability can technically be any number \in [0, 1].
  • However, this prior assumes that \pi has a discrete set of possibilities: 20%, 50%, or 80%.

Example 3: Building a Model

• In the second step of our analysis, we collect and process data which can inform our understanding of \pi.

• Here, Y = the number of the six games in the 1997 re-match that Kasparov wins.

  • As chess match outcome isn’t predetermined, Y is a random variable that can take any value in \{0, 1, 2, 3, 4, 5, 6\}.

• Note that Y inherently depends upon \pi.

  • If \pi = 0.80, Y would also be high (on average).
  • If \pi = 0.20, Y would also be low (on average).

• Thus, we must model this dependence of Y on \pi using a conditional probability model.

Binomial Data Model

• We must make two assumptions about the chess match:
  • Games are independent (the outcome of one game does not influence the outcome of another).
  • Kasparov has an equal probability of winning any game in the match.
    • i.e., probability of winning does not increase or decrease as the match goes on.
• We will use a binomial model for this problem.
  • In our case,

Y|\pi \sim \text{Bin}(6, \pi)

Binomial Data Model

• Let’s assume \pi = 0.8.

• The probability that he would win all 6 games is approximately 26%.

f(y=6|\pi=0.8) = {6 \choose 6} 0.8^6 (1-0.8)^{6-6},

dbinom(6, 6, 0.8)

[1] 0.262144

Binomial Data Model

• Let’s assume \pi = 0.8.

• The probability that he would win none of the games is approximately 0%.

f(y=0|\pi=0.8) = {6 \choose 0} 0.8^0 (1-0.8)^{6-0},

dbinom(0, 6, 0.8)

[1] 6.4e-05

Binomial Data Model

• Visually,

Binomial Data Model

• Note that the Binomial gives us the theoretical model of the data we might observe.

  • Kasparov only won one of the six games against Deep Blue in 1997 (Y=1).

• Next, we determine how compatible this particular data is with the various possible values of \pi.

  • What is the likelihood of Kasparov winning Y=1 game under each possible \pi?

• Recall, f(y|\pi) = L(\pi|Y=y). When Y=1,

\begin{align*} L(\pi | y = 1) &= f(y=1|\pi) \\ &= {6 \choose 1} \pi^1 (1-\pi)^{6-1} \\ &= 6\pi(1-\pi)^5 \end{align*}

• Note that we do not expect all likelihoods to sum to 1.

Binomial Data Model

• Use your results from earlier to tell me the resulting likelihood values.

Binomial Data Model

• Use your results from earlier to tell me the resulting likelihood values.

• As we can see, the likelihoods do not sum to 1.

Normalizing Constant

• Bayes’ Rule requires three pieces of information:
  • Prior
  • Likelihood
  • Normalizing constant
• Normalizing constant: ensures that the sum of all probabilities is equal to 1.
  • It can be a scalar or a function.
  • Every probability distribution that does not sum to 1 will have a normalizing constant.

Normalizing Constant

• We now must determine the total probability that Kasparov would win Y=1 games across all possible win probabilities \pi, f(y=1).

\begin{align*} f(y=1) =& \sum_{\pi} L(\pi |y=1)f(\pi) \\ =& L(\pi=0.2|y=1)f(\pi=0.2) + L(\pi=0.5|y=1)f(\pi=0.5) + \\ & L(\pi=0.8|y=1)f(\pi=0.8) \end{align*}

• Work with your group to find the normalizing constant.

Normalizing Constant

• We now must determine the total probability that Kasparov would win Y=1 games across all possible win probabilities \pi, f(y=1).

\begin{align*} f(y=1) =& \sum_{\pi} L(\pi |y=1)f(\pi) \\ =& L(\pi=0.2|y=1)f(\pi=0.2) + L(\pi=0.5|y=1)f(\pi=0.5) + \\ & L(\pi=0.8|y=1)f(\pi=0.8) \\ \approx& 0.3932 \cdot 0.10 + 0.0938 \cdot 0.25 + 0.0015 \cdot 0.65 \\ \approx& 0.0637 \end{align*}

• Across all possible values of \pi, there is about a 6% chance that Kasparov would have won only one game.

Posterior Probability Model

• Now recall,

\text{posterior} = \frac{\text{prior} \times \text{likelihood}}{\text{normalizing constant}}

• In our example, where y = 1,

f(\pi | y=1) = \frac{f(\pi) L(\pi | y = 1)}{f(y=1)} \ \text{for} \ \pi \in \{ 0.2, 0.5, 0.8\}

• Work with your group to find the posterior probabilities.

  • You will have one posterior probability for each value of \pi.

Posterior Probability Model

• Note!! We do not have to calculate the normalizing constant!

• We can treat f(Y=y) as a constant with respect to \pi.

• Then, we say that

\begin{align*} f(\pi | y) &= \frac{f(\pi) L(\pi|y)}{f(y)} \\ & \propto f(\pi) L(\pi|y) \\ \\ \text{posterior} &\propto \text{prior} \cdot \text{likelihood} \end{align*}

Wrap Up

• Today we learned how to, in general, approach Bayesian analysis.

• You now are ready to work on Assignment 2: Thinking Like a Bayesian.

  • .qmd file is available to download on Canvas.

• Next week: conjugate families and one sample inference

  • Beta-binomial
  • Gamma-Poisson
  • Normal Normal