IM construction

This page shows a toy example of how to construct a possibilistic inferential model (IM) when the data is assumed to come from a Binomial distribution with \(n\) trials and probability of success \(\theta\).

Relative likelihood

To construct an inferential model, we first construct the relative likelihood: \[ R(x, \theta) = \frac{p_{\theta}(x)}{\sup_{\vartheta} p_\vartheta(x)}, \] where \(p_{\theta}(x)\) is the likelihood function of Bin(\(n, \theta\)):

\[ p_\theta(x) = \begin{pmatrix} n \\ x \end{pmatrix} \theta^x (1-\theta)^{n-x}, \quad x = 0, 1, \cdots, n, \quad \theta \in [0, 1]. \] The maximum likelihood estimator of a Binomial distribution is \(\frac{x}{n}\). Thus the relative likelihood is:

\[R(x, \theta) = \frac{p_\theta(x)}{p_{\widehat{\theta}}(x)} = \left( \frac{n\theta}{x}\right)^x \left( \frac{n - n\theta}{n-x}\right)^{n-x} \]

Contour function

The next step is a validation step:

\[\pi_x(\theta) = P_\theta(R(X, \theta) \le R(x, \theta)).\]

For the Binomial distribution, there is a closed form for the contour.

\[ \pi_x(\theta) = \sum_{y=0}^n 1 \{ R(y, \theta) \le R(x, \theta) \} p_\theta(y). \]

The code below finds the contour function \(\pi_x(\theta)\) given observation \(x\) and the parameter of interest \(\theta\).

R_binomial <- function(x, n, theta){
  
  R <- (n*theta/x)^x * (n*(1-theta)/(n-x))^(n-x)
  return(R)
  
}

# closed form for x=n
contour_binomial <- function(theta, n, x){
  
  RHS <- R_binomial(x, n, theta)
  total <- 0
  for (i in 0:n){
    total <- total + (1*(R_binomial(i, n, theta) <= RHS)) * dbinom(i, n, theta)
  }
  return(total)
  
}

Example demonstration

Now assume we observe \(6\) successes from \(10\) trilas, let’s plot the contour function.

n <- 10
x <- 6

theta <- seq(0.01, .99, by = .005)

pi_binomial <- contour_binomial(theta, n, x)
plot(theta, pi_binomial, type = "l", xlab = expression(theta), ylab = expression(pi[x](theta)))
abline(h=.05, lty=2)
segments(0.3, 0, 0.5, 0, col="red", lwd=2)
segments(0.5, 0, 0.5, contour_binomial(0.5, n, x), lty=3, col="red")
points(0.5, contour_binomial(0.5, n, x), col="red", pch=19)

As shown above, the black curve is the contour function for \(x=6\). The possibility measure (or upper probability) of the hypothesis \(h=[0.3,0.5]\) can be found by maximization:

\[ \Pi_x(h) = \sup_{\theta \in [0.3,0.5]} \pi_x(\theta) \approx 0.754,\] where the red circle is.

To find a \(95\%\) confidence set, the black dashed line at \(0.05\) shows that the set is approximately \([0.25, 0.85]\).

Binomial

Shih-Ni Prim

2026-09-01

IM construction

Relative likelihood

Contour function

Example demonstration

Reference