sample_aggregator {braggR}

Revealed Aggregator

Description

This function allows the user to compute the revealed aggregator from Satopää, V.A. (2021): Regularized Aggregation of One-off Probability Predictions. The current version of the paper is available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3769945.

Usage

sample_aggregator(
  p,
  p0 = NULL,
  alpha = NULL,
  beta = NULL,
  a = 1/2,
  b = 1/2,
  num_sample = 1e+06,
  burnin = num_sample/2,
  thin = 1,
  seed = 1
)

Arguments

p

Vector of K \ge 2 forecasters' probability estimates of a future binary event. These values represent probability predictions and must be strictly between 0 and 1.

p0

The forecasters' common prior. This represents a probability prediction based on some of the forecasters' common evidence and must be strictly between 0 and 1.

alpha, beta

The shape and scale parameters of the prior beta distribution of the common prior. If omitted, the sampler uses the fixed common prior given by p0. However, if alpha and beta are provided, they must be strictly positive. In this case, the common prior p0 will be treated as a random variable and sampled along with the other model parameters.

a, b

The parameters for the prior distribution of (\rho, \gamma, \delta) in Satopää, V.A. (2021). The default choice a = 1/2 and b = 1/2 gives the Jeffreys' independence prior. If p_0 is not equal to 0.5, then a = 1 and b = 1/2 give the Jeffreys' prior.

num_sample

The number of posterior samples to be drawn. This does not take into account burnin and thinning.

burnin

The number of the initial num_sample posterior draws that are discarded for burnin. This value cannot exceed num_sample.

thin

After burnin draws have been discarded, the final sample is formed by keeping every thin'th value. To ensure that the final sample holds at least two draws, thin can be at most (num_sample-burnin)/2.

seed

The seed value for random value generation.

Value

A data frame with rows representing posterior draws of (p*, \rho, \gamma, \delta, p0). The columns are:

• aggregate: The posterior samples of the oracle aggregator p*. The average of these values gives the revealed aggregator p''. The 95% interval of these values gives the 95% credible interval of the oracle aggregator.

• rho: The posterior samples of the forecasters' shared evidence, \rho.

• gamma: The posterior samples of the forecasters' total evidence, \gamma. The difference gamma-rho gives the posterior samples of the forecasters' rational disagreement.

• delta: The posterior samples of the forecasters' total evidence plus noise, \delta. The difference delta-gamma gives the posterior samples of the forecasters' irrational disagreement.

• p0: The posterior samples of the forecasters' common prior. If a beta prior distribution is not specified via the arguments alpha and beta, then all elements of this column are equal to the fixed common prior given by the p0 argument.

Examples

# Illustration on Scenario B in Satopää, V.A. (2021).
# Forecasters' probability predictions:
p = c(1/2, 5/16, 1/8, 1/4, 1/2)

# Aggregate with a fixed common prior of 0.5.
# Sample the posterior distribution:
post_sample = sample_aggregator(p, p0 = 0.5, num_sample = 10^6, seed = 1)
# The posterior means of the model parameters:
colMeans(post_sample[,-1])
# The posterior mean of the oracle aggregator, a.k.a., the revealed aggregator:
mean(post_sample[,1])
# The 95% credible interval for the oracle aggregator:
quantile(post_sample[,1], c(0.025, 0.975))


# Aggregate based a uniform distribution on the common prior
# Recall that Beta(1,1) corresponds to the uniform distribution.
# Sample the posterior distribution:
post_sample = sample_aggregator(p, alpha = 1, beta = 1, num_sample = 10^6, seed = 1)
# The posterior means of the oracle aggregate and the model parameters:
colMeans(post_sample)