Bayesian Methods

Introduction

Bayesian inference provides a principled framework for updating beliefs as new evidence arrives. In project risk management, this means starting with prior estimates of risk probability (based on historical data or expert judgment), then refining those estimates as on-site observations accumulate.

Bayes’ theorem states:

\[ P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)} \]

The PRA package provides four Bayesian functions organized into two stages:

library(PRA)

Step 1: Prior Risk Probability

Before any observations are made, risk_prob() computes the probability of a risk event R occurring given two potential root causes. For each cause, we supply:

• cause_probs — prior probability that the cause is present
• risks_given_causes — P(R | cause present)
• risks_given_not_causes — P(R | cause absent)

cause_probs <- c(0.3, 0.2)
risks_given_causes <- c(0.8, 0.6)
risks_given_not_causes <- c(0.2, 0.4)

prior_risk <- risk_prob(cause_probs, risks_given_causes, risks_given_not_causes)
cat("Prior probability of risk event R:", round(prior_risk, 3), "\n")

Prior probability of risk event R: 0.82

Step 2: Prior Cost Distribution

Given the prior risk probabilities, cost_pdf() samples the cost distribution before any field observations. Three independent risk events can each contribute cost if they occur.

risk_probs <- c(0.3, 0.5, 0.2)
means_given_risks <- c(10000, 15000, 5000)
sds_given_risks <- c(2000, 1000, 1000)
base_cost <- 2000

prior_samples <- cost_pdf(
  num_sims          = 5000,
  risk_probs        = risk_probs,
  means_given_risks = means_given_risks,
  sds_given_risks   = sds_given_risks,
  base_cost         = base_cost
)

We will compare this to the posterior distribution in Step 4.

Step 3: Posterior Risk Probability (Bayesian Update)

After inspecting the project site, we observe that Cause 1 is present (= 1). Cause 2 has not yet been assessed (= NA). risk_post_prob() updates the risk probability using only the available evidence — NA causes are treated as unobserved and do not contribute to the update.

# C1 observed as present; C2 not yet assessed
observed_causes <- c(1, NA)

posterior_risk <- risk_post_prob(
  cause_probs, risks_given_causes,
  risks_given_not_causes, observed_causes
)
cat("Posterior probability of risk event R:", round(posterior_risk, 3), "\n")

Posterior probability of risk event R: 0.632

Observing Cause 1 (which has a strong link to R) raises the risk probability substantially. The NA for Cause 2 is simply ignored — only confirmed observations drive the update.

prob_data <- data.frame(
  Stage       = c("Prior", "Posterior"),
  Probability = c(prior_risk, posterior_risk)
)

p <- ggplot2::ggplot(prob_data, ggplot2::aes(x = Stage, y = Probability, fill = Stage)) +
  ggplot2::geom_col(width = 0.5, show.legend = FALSE) +
  ggplot2::geom_text(ggplot2::aes(label = round(Probability, 3)),
    vjust = -0.4, size = 4.5
  ) +
  ggplot2::scale_fill_manual(values = c("Prior" = "steelblue", "Posterior" = "tomato")) +
  ggplot2::scale_y_continuous(limits = c(0, 1), labels = scales::percent) +
  ggplot2::labs(
    title = "Bayesian Update: Risk Probability",
    x     = NULL,
    y     = "P(Risk Event R)"
  ) +
  ggplot2::theme_minimal(base_size = 13)

print(p)

Step 4: Posterior Cost Distribution

Now that we know Cause 1 is present (Risk 1 occurs), and one risk remains unobserved (Risk 2 = NA), cost_post_pdf() samples the posterior cost distribution. Observed risks that occurred (= 1) add their cost; unobserved risks (= NA) are excluded from the simulation.

observed_risks <- c(1, NA, 1) # Risk 1 and 3 confirmed; Risk 2 not yet assessed

posterior_samples <- cost_post_pdf(
  num_sims          = 5000,
  observed_risks    = observed_risks,
  means_given_risks = means_given_risks,
  sds_given_risks   = sds_given_risks,
  base_cost         = base_cost
)

xlim_range <- range(c(prior_samples, posterior_samples))

# Prior cost histogram
hist(prior_samples,
  breaks = 40, freq = FALSE,
  col = rgb(0.27, 0.51, 0.71, 0.5), # steelblue, semi-transparent
  border = "white",
  xlim = xlim_range,
  main = "Prior vs. Posterior Cost Distribution",
  xlab = "Total Cost ($)",
  ylab = "Density"
)

# Posterior cost histogram (overlaid)
hist(posterior_samples,
  breaks = 40, freq = FALSE,
  col = rgb(0.84, 0.24, 0.31, 0.5), # tomato, semi-transparent
  border = "white",
  add = TRUE
)

abline(v = mean(prior_samples), col = "steelblue", lty = 2, lwd = 2)
abline(v = mean(posterior_samples), col = "tomato", lty = 2, lwd = 2)

legend("topright",
  legend = c(
    paste0("Prior  (mean = $", format(round(mean(prior_samples)), big.mark = ",")),
    paste0("Posterior (mean = $", format(round(mean(posterior_samples)), big.mark = ","))
  ),
  fill = c(rgb(0.27, 0.51, 0.71, 0.5), rgb(0.84, 0.24, 0.31, 0.5)),
  bty = "n"
)

Summary

The Bayesian workflow in PRA follows a natural before-and-after structure:

1. Before observations — use risk_prob() and cost_pdf() to characterize the risk landscape.
2. As evidence arrives — use risk_post_prob() and cost_post_pdf() to update estimates.
3. NA values in observed_causes and observed_risks represent causes/risks that have not yet been assessed; they are correctly excluded from the Bayesian update.

This approach is particularly powerful in phased projects where information about risk drivers becomes available progressively, allowing cost forecasts to be refined at each stage.