1.2.1 Bayes Rule

Bayesian inference takes into account:

• the consistency of the evidence with a hypothesis
• the uniqueness of the evidence to that hypothesis
• background knowledge about the problem.

1.2.2 Illustration 1

I draw a card from a deck and ask What are the chances it is a Jack of Spades?

• Just 1 in 52.

Now I tell you that the card is indeed a spade. What would you guess?

• 1 in 13

What if I told you it was a heart?

• No chance it is the Jack of Spades

What if I said it was a face card and a spade.

• 1 in 3.

1.2.3 Illustration 1

These answers are applications of Bayes’ rule.

In each case the answer is derived by assessing what is possible, given the new information, and then assessing how likely the outcome of interest among the states that are possible. In all the cases you calculate:

\[\text{Prob Jack of Spades | Info} = \frac{\text{Is Jack of Spades Consistent w/ Info?}}{\text{How many cards are consistent w/ Info?}} \]

1.2.4 Illustration 2 Interpreting Your Test Results

You take a test to see whether you suffer from a disease that affects 1 in 100 people. The test is good in the following sense:

• if you have the disease, then with a 99% probability it will say you have the disease
• if you do not have it, then with a 99% probability, it will say that you do not have it

The test result says that you have the disease. What are the chances you have it?

1.2.5 Illustration 2 Interpreting Your Test Results

• It is not 99%. 99% is the probability of the result given the disease, but we want the probability of the disease given the result.

• The right answer is 50%, which you can think of as the share of people that have the disease among all those that test positive. For example

• e.g. if there were 10,000 people, then 100 would have the disease and 99 of these would test positive. But 9,900 would not have the disease and 99 of these would test positive. So the people with the disease that test positive are half of the total number testing positive.

1.2.6 Illustration 2. A picture

What’s the probability of being a circle given you are black?

1.2.7 Illustration 2. More formally.

As an equation this might be written:

\[\text{Prob You have the Disease | Pos} = \frac{\text{How many have the disease and test pos?}}{\text{How many people test pos?}}\]

1.2.8 Two Child Problem

Consider last an old puzzle described in Gardner (1961).

• Mr Smith has two children, \(A\) and \(B\).
• At least one of them is a boy.
• What are the chances they are both boys?

To be explicit about the puzzle, we will assume that the information that one child is a boy is given as a truthful answer to the question “is at least one of the children a boy?”

Assuming also that there is a 50% probability that a given child is a boy.

1.2.9 Two Child Problem

As an equation:

\[\text{Prob both boys | Not both girls} = \frac{\text{Prob both boys}}{\text{Prob not both girls}} = \frac{\text{1 in 4}}{\text{3 in 4}}\]

1.2.11 Bayes Rule

Formally, all of these equations are applications of Bayes’ rule which is a simple and powerful formula for deriving updated beliefs from new data.

The formula is given as:

\[\Pr(H|\mathcal{D})=\frac{\Pr(\mathcal{D}|H)\Pr(H)}{\Pr(\mathcal{D})}\\ =\frac{\Pr(\mathcal{D}|H)\Pr(H)}{\sum_{H'}\Pr(\mathcal{D}|H')\Pr(H'))}\]

1.2.13 Useful Distributions: Beta and Dirichlet Distributions

• Bayes rule requires the ability to express a prior distribution but it does not require that the prior have any particular properties other than being probability distributions.
• Sometimes however it can be useful to make use of “off the shelf” distributions.

Consider the share of people in a population that voted. This is a quantity between 0 and 1.

1.2.14 Useful Distributions: Beta and Dirichlet Distributions

• Two people might may both believe that the turnout was around 50% but differ in how certain they are about this claim.
• One might claim to have no information and to believe any turnout rate between 0 and 100% is equally likely; another might be completely confident that the number if 50%.

Here the parameter of interest is a share. The Beta and Dirichlet distributions are particularly useful for representing beliefs on shares.

1.2.15 Beta

• The Beta distribution is a distribution over the \([0,1]\) that is governed by two parameters, \(\alpha\) and \(\beta\).
• In the case in which both \(\alpha\) and \(\beta\) are 1, the distribution is uniform – all values are seen as equally likely.
• As \(\alpha\) rises large outcomes are seen as more likely
• As \(\beta\) rises, lower outcomes are seen as more likely.
• If both rise proportionately the expected outcome does not change but the distribution becomes tighter.

An attractive feature is that if one has a prior Beta(\(\alpha\), \(\beta\)) over the probability of some event, and then one observes a positive case, the Bayesian posterior distribution is also a Beta with with parameters \(\alpha+1, \beta\). Thus if people start with uniform priors and build up knowledge on seeing outcomes, their posterior beliefs should be Beta.

1.2.16 Beta

Here is a set of such distributions.

Beta distributions

1.2.17 Dirichlet distributions.

The Dirichlet distributions are generalizations of the Beta to the situation in which there are beliefs not just over a proportion, or a probability, but over collections of probabilities.

• If four outcomes are possible and each is likely to occur with probability \(p_k\), \(k=1,2,3,4\) then beliefs are distributions over a three dimensional unit simplex.

• The distribution has as many parameters as there are outcomes and these are traditionally recorded in a vector, \(\alpha\).

• As with the Beta distribution, an uninformative prior (Jeffrey’s prior) has \(\alpha\) parameters of \((.5,.5,.5, \dots)\) and a uniform (“flat”) distribution has \(\alpha = (1,1,1,,\dots)\).

• The Dirichlet updates in a simple way. If you have a Dirichlet prior with parameter \(\alpha = (\alpha_1, \alpha_2, \dots)\) and you observe outcome \(1\), for example, then then posterior distribution is also Dirichlet with parameter vector \(\alpha' = (\alpha_1+1, \alpha_2,\dots)\).

1.4.8 Simple model: Now Let’s Run It

M <- stan(file = "assets/one_var.stan", 
          data = some_data)

When you run the model you get a lot of useful output on the estimation and the posterior distribution. Here though are the key results:

These look good.

The Rhat at the end tells you about convergence. You want this very close to 1.

1.4.11 Building up

Let’s go back to the code.

There we had three key blocks: data, parameters, and model

More generally the blocks you can specify are:

• data (define the vars that will be coming in from the data list)
• transformed data (can be used for preprocessing)
• parameters (required: defines the parameters to be estimated)
• transformed parameters (transformations of parameters useful for computational reasons and sometimes for clarity)
• model (give priors and likelihood)
• generated quantities (can be used for post processing)

1.4.19 A multilevel model

Here is a model for this

ml_model <- '
data {
  vector[100] Y;
  int<lower=0,upper=1> X[100];
  int village[100];
}
parameters {
  vector<lower=0>[3] sigma; 
  vector[10] a;
  vector[10] b;
  real mu_a;
  real mu_b;
}
transformed parameters {
  vector[100] Y_vx;
  for (i in 1:100) Y_vx[i] = a[village[i]] + b[village[i]] * X[i];
}
model {
  a ~ normal(mu_a, sigma[1]);
  b ~ normal(mu_b, sigma[2]);
  Y ~ normal(Y_vx, sigma[3]);
}
'

1.4.20 A multilevel model

Here is a slightly more general version: https://github.com/stan-dev/example-models/blob/master/ARM/Ch.17/17.1_radon_vary_inter_slope.stan

1.5.1 A game and a structural model

Say that a set of people in a population are playing sequential prisoner’s dilemmas.

In such games selfish behavior might suggest defections by everyone everywhere. But of course people often cooperate. Why might this be?

• One possible reason is that some people are irrational, in the sense that they simply choose to cooperate, ignoring the payoffs.
• Another possibility is that rational people think that others are irrational, in the sense that they think that others will reciprocate when they observe cooperative action

1.5.2 Model

We will capture some of this intuition with a behavioral type model in which

• each player has a “rationality” propensity of \(r_i\) – this is the probability with which they choose to do the rational thing, rather than the generous thing
• \(r_i \sim U[0, \theta]\) for \(\theta > .5\).
• A player with rationality propensity of \(r_i\) believes \(r_j \sim [0, r_i]\). So everyone assumes that they are the most rational people in the room…
• The game is such that:
• second mover: a second mover with rationality propensity \(r_i\) will cooperate with probability \(1-r_i\) if the first mover cooperated; otherwise they defect
• first mover: a first mover with \(r_i\) will cooperate nonstrategically with probability \((1-r_i)\); however with probability \(r_i\) they will also cooperate strategically if they think that the second mover has \(r_j<.25\).

1.5.4 Event Probabilities

What then are the probabilities of each of the possible outcomes?

• There will be cooperation by both players with probability \((\int_0^{.5} p(r_i) dr_i + \int_{.5}^1 p(r_i)(1-r_i) dr_i)\int_0^1p(r_i)(1-r_i)dr_i\)
• There will be cooperation by player 1 only with probability \((\int_0^{.5} p(r_i) dr_i + \int_{.5}^1 p(r_i)(1-r_i) dr_i)(\int_0^1p(r_i)(r_i)dr_i)\)
• There will be cooperation by neither with probability: \(1-\int_0^{.5} p(r_i) dr_i - \int_{.5}^1 p(r_i)(1-r_i) dr_i\)

where \(p\) is the density function on \(r_i\) given \(\theta\)

1.5.5 Event probabilities

Given the assumption on \(p\)

• There will be cooperation by both players with probability \((1+.25/\theta -.5\theta)(1-.5\theta)\)
• There will be cooperation by player 1 only with probability \((1+.25/\theta -.5\theta)(.5\theta)\)
• There will be cooperation by neither player with probability \((.5\theta-.25/\theta)\)

1.5.6 Data

• We have data on the actions of the first movers and the second movers and are interested in the distribution of the \(p_i\)s.

• Lets collapse that data into a simple list of the number of each type of game outcome:

• And say we start off with a uniform prior of \(\theta\).

• What should we conclude about \(\theta\)?

1.6.2 Illustration: “Lipids” data

data("lipids_data")

lipids_data |> kable()

Note that in compact form we simply record the number of units (“count”) that display each possible pattern of outcomes on the three variables (“event”).[^1]