Bayesian vs Frequentist

Same data, different questions

Bayesian and frequentist statistics look at the same data but ask different questions:

	Frequentist	Bayesian
Parameters are…	Fixed but unknown	Random variables with distributions
Probability means…	Long-run frequency	Degree of belief
Result	Point estimate + confidence interval	Full posterior distribution
“There’s a 95% chance…”	…that this procedure captures the true value	…that the true value is in this interval

The frequentist says: “If I repeated this experiment forever, 95% of my CIs would contain the true value.” The Bayesian says: “Given what I’ve seen, I’m 95% sure the true value is in this range.”

Most people actually think like Bayesians (“what’s the probability the parameter is between A and B?”) but compute like frequentists (p-values, CIs).

Side-by-side comparison

The simulation below runs the same experiment and shows both the frequentist confidence interval and the Bayesian credible interval. Watch how they differ — especially with small samples and informative priors.

#| standalone: true
#| viewerHeight: 620

library(shiny)

ui <- fluidPage(
  tags$head(tags$style(HTML("
    .stats-box {
      background: #f0f4f8; border-radius: 6px; padding: 14px;
      margin-top: 12px; font-size: 14px; line-height: 1.9;
    }
    .stats-box b { color: #2c3e50; }
  "))),

  sidebarLayout(
    sidebarPanel(
      width = 3,

      sliderInput("true_mu", HTML("True &mu;:"),
                  min = -3, max = 3, value = 1, step = 0.5),

      sliderInput("n", "Sample size:",
                  min = 2, max = 200, value = 10, step = 1),

      sliderInput("prior_mu", "Bayesian prior mean:",
                  min = -3, max = 3, value = 0, step = 0.5),

      sliderInput("prior_sd", "Prior SD:",
                  min = 0.5, max = 10, value = 2, step = 0.5),

      actionButton("go", "New experiment", class = "btn-primary", width = "100%"),

      uiOutput("results")
    ),

    mainPanel(
      width = 9,
      fluidRow(
        column(6, plotOutput("interval_plot", height = "420px")),
        column(6, plotOutput("repeat_plot",   height = "420px"))
      )
    )
  )
)

server <- function(input, output, session) {

  dat <- reactive({
    input$go
    true_mu  <- input$true_mu
    n        <- input$n
    prior_mu <- input$prior_mu
    prior_sd <- input$prior_sd
    sigma    <- 2

    y <- rnorm(n, mean = true_mu, sd = sigma)
    y_bar <- mean(y)
    se <- sigma / sqrt(n)

    # Frequentist 95% CI
    freq_lo <- y_bar - 1.96 * se
    freq_hi <- y_bar + 1.96 * se

    # Bayesian posterior
    prior_prec <- 1 / prior_sd^2
    data_prec  <- n / sigma^2
    post_prec  <- prior_prec + data_prec
    post_sd    <- 1 / sqrt(post_prec)
    post_mu    <- (prior_prec * prior_mu + data_prec * y_bar) / post_prec

    bayes_lo <- qnorm(0.025, post_mu, post_sd)
    bayes_hi <- qnorm(0.975, post_mu, post_sd)

    # Repeated experiments for right panel
    k <- 50
    reps <- t(replicate(k, {
      yy <- rnorm(n, mean = true_mu, sd = sigma)
      yy_bar <- mean(yy)
      f_lo <- yy_bar - 1.96 * se
      f_hi <- yy_bar + 1.96 * se

      d_prec <- n / sigma^2
      p_prec <- prior_prec + d_prec
      p_sd <- 1 / sqrt(p_prec)
      p_mu <- (prior_prec * prior_mu + d_prec * yy_bar) / p_prec
      b_lo <- qnorm(0.025, p_mu, p_sd)
      b_hi <- qnorm(0.975, p_mu, p_sd)

      c(yy_bar, f_lo, f_hi, p_mu, b_lo, b_hi)
    }))

    list(true_mu = true_mu, y_bar = y_bar,
         freq_lo = freq_lo, freq_hi = freq_hi,
         post_mu = post_mu, post_sd = post_sd,
         bayes_lo = bayes_lo, bayes_hi = bayes_hi,
         reps = reps, prior_mu = prior_mu)
  })

  output$interval_plot <- renderPlot({
    d <- dat()

    par(mar = c(4.5, 8, 3, 1))
    xlim <- range(c(d$freq_lo, d$freq_hi, d$bayes_lo, d$bayes_hi, d$true_mu)) +
            c(-0.5, 0.5)

    plot(NULL, xlim = xlim, ylim = c(0.5, 2.5),
         yaxt = "n", ylab = "", xlab = expression(mu),
         main = "This Experiment")
    axis(2, at = 1:2, labels = c("Frequentist\n95% CI", "Bayesian\n95% CrI"),
         las = 1, cex.axis = 0.85)

    # Frequentist
    segments(d$freq_lo, 1, d$freq_hi, 1, lwd = 4, col = "#e74c3c")
    points(d$y_bar, 1, pch = 19, cex = 1.5, col = "#e74c3c")

    # Bayesian
    segments(d$bayes_lo, 2, d$bayes_hi, 2, lwd = 4, col = "#3498db")
    points(d$post_mu, 2, pch = 19, cex = 1.5, col = "#3498db")

    # True value
    abline(v = d$true_mu, lty = 2, lwd = 2, col = "#2c3e50")
    text(d$true_mu, 2.4, expression("True " * mu), cex = 0.9, col = "#2c3e50")
  })

  output$repeat_plot <- renderPlot({
    d <- dat()
    k <- nrow(d$reps)

    par(mar = c(4.5, 4, 3, 1))

    freq_covers <- d$reps[, 2] <= d$true_mu & d$reps[, 3] >= d$true_mu
    bayes_covers <- d$reps[, 5] <= d$true_mu & d$reps[, 6] >= d$true_mu

    xlim <- range(d$reps[, 2:6], d$true_mu) + c(-0.5, 0.5)

    plot(NULL, xlim = xlim, ylim = c(1, k),
         xlab = expression(mu), ylab = "Experiment #",
         main = paste0(k, " repeated experiments"))

    for (i in seq_len(k)) {
      # Frequentist (left-shifted slightly)
      clr_f <- if (freq_covers[i]) "#e74c3c" else "#e74c3c40"
      segments(d$reps[i, 2], i - 0.15, d$reps[i, 3], i - 0.15,
               lwd = 1.5, col = clr_f)

      # Bayesian (right-shifted slightly)
      clr_b <- if (bayes_covers[i]) "#3498db" else "#3498db40"
      segments(d$reps[i, 5], i + 0.15, d$reps[i, 6], i + 0.15,
               lwd = 1.5, col = clr_b)
    }

    abline(v = d$true_mu, lty = 2, lwd = 2, col = "#2c3e50")

    legend("topright", bty = "n", cex = 0.8,
           legend = c(
             paste0("Freq CI (", sum(freq_covers), "/", k, " cover)"),
             paste0("Bayes CrI (", sum(bayes_covers), "/", k, " cover)")
           ),
           col = c("#e74c3c", "#3498db"), lwd = 3)
  })

  output$results <- renderUI({
    d <- dat()
    tags$div(class = "stats-box",
      HTML(paste0(
        "<b>Frequentist:</b><br>",
        "Estimate: ", round(d$y_bar, 3), "<br>",
        "95% CI: [", round(d$freq_lo, 3), ", ", round(d$freq_hi, 3), "]<br>",
        "<hr style='margin:8px 0'>",
        "<b>Bayesian:</b><br>",
        "Posterior mean: ", round(d$post_mu, 3), "<br>",
        "95% CrI: [", round(d$bayes_lo, 3), ", ", round(d$bayes_hi, 3), "]<br>",
        "<hr style='margin:8px 0'>",
        "<small>CrI is narrower because the prior adds information.</small>"
      ))
    )
  })
}

shinyApp(ui, server)

Things to try

• n = 5, prior centered at 0, true mu = 1: the Bayesian CrI is narrower but pulled toward 0 (shrinkage). The frequentist CI is wider but centered on the data.
• n = 200: both intervals are nearly identical. With lots of data, the prior washes out and Bayesian = frequentist.
• Set a wrong prior (prior mean = -3, true mu = 2, n = 5): the Bayesian interval gets pulled toward -3. A bad prior hurts with small samples. Slide n up — the data corrects it.
• Right panel: the frequentist CI is designed so that ~95% of the red intervals cover the truth across repetitions. The Bayesian CrI coverage depends on how good the prior is.

The bottom line

Use frequentist when…	Use Bayesian when…
You want procedure guarantees (coverage)	You want direct probability statements
You have no prior information	You have real prior knowledge
Regulatory/peer review expects it	Small samples, need to borrow strength
Simple problems	Complex hierarchical models

In practice, most applied researchers use frequentist methods but interpret them like Bayesians. Understanding both helps you know what your numbers actually mean.