Code for blog post on hindsight bias

hindsight_bias_122225.R

# Setup -------------------------------------------------------------------

set.seed(223)
library(tidyverse)
theme_set(theme_classic())

# Compute expected absolute error of probability estimate
abs_error <- function(x) {
  2 * x * (1 - x)
}

# Compute expected guess with memory noise (without knowledge of answer)
expected_guess_from_memory <- function(m, noise, tol = 0.01) {
  logit_vals <- seq(-10, 10, tol)
  bayes_numerator <- dnorm(logit_vals, qlogis(m), noise) * dlogis(logit_vals)
  as.numeric((bayes_numerator / sum(bayes_numerator)) %*% plogis(logit_vals))
}

# Compute expected guess with memory noise + knowledge that answer is FALSE (0)
# (Same as expected error when all statements resolve to FALSE)
expected_guess_with_answer_zero <- function(m, noise, tol = 0.01) {
  logit_vals <- seq(-10, 10, tol)
  bayes_numerator <- dnorm(logit_vals, qlogis(m), noise) * dlogis(logit_vals) * (1 - plogis(logit_vals))
  as.numeric((bayes_numerator / sum(bayes_numerator)) %*% plogis(logit_vals))
}

# Compute expected error with memory noise (without knowledge of answer)
expected_error_from_memory <- function(m, noise, tol = 0.01) {
  logit_vals <- seq(-10, 10, tol)
  bayes_numerator <- dnorm(logit_vals, qlogis(m), noise) * dlogis(logit_vals)
  as.numeric(
    (bayes_numerator / sum(bayes_numerator)) %*% abs_error(plogis(logit_vals))
  )
}


# Main Simulation ---------------------------------------------------------

# Number of samples
N <- 100000

# Sample binary truth values of statements, with 50% chance of TRUE
T <- sample(c(0, 1), N, replace = TRUE)

# Sample recorded probability estimates based on beta distribution
X <- rbeta(N, 1 + T, 2 - T)

# ... this transforms all estimates to be relative to T = FALSE
X_mirrored <- if_else(T == 1, 1 - X, X)

# Std. deviation of Gaussian memory noise in log-odds
noise <- 0.5

# Sample memory signals from X, with added noise
M <- plogis(rnorm(N, qlogis(X), noise))

# ... this transforms remembered estimates to be relative to T = FALSE
M_mirrored <- if_else(T == 1, 1 - M, M)

# Compute Bayesian estimates from memory without knowledge of answer
M_corrected <- map_dbl(seq_along(M), \(i) expected_guess_from_memory(M[i], noise))

# Compute Bayesian estimates from memory with knowledge of answer (where T = FALSE)
M_with_answer <- map_dbl(seq_along(M), \(i) expected_guess_with_answer_zero(M_mirrored[i], noise))

# Compute expected error from memory without knowledge of answer
M_exp_error <- map_dbl(seq_len(length(M)), \(i) expected_error_from_memory(M[i], noise))


# Expectations of error (all approximately equal by law of iterated expectations):
mean(abs_error(X)) # = mean error in initial judgments
mean(M_exp_error) # = expected error from memory without answer
mean(M_with_answer) # = expected error from memory with answer