IM construction

This page shows a toy example of how to construct a possibilistic inferential model (IM).

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 with parameter \(\theta\) and observed data \(x\).

Assuming the data \(X \sim N(\theta, 1)\) and the sample size \(n=1\),

\[ \begin{align} R(x,\theta) &= \frac{p_\theta(x)}{p_{\hat{\theta}}(x)} = p_\theta(x) = p(|x-\theta|), \end{align} \] where \(p\) is the density function for standard Normal distribution.

Contour function

The next step is a validation step:

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

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

\[ \begin{align} \pi_x(\theta) &= P_\theta(p(X-\theta) \le p(x-\theta)) \\ &= P_\theta(|X-\theta| \le |x-\theta|) \\ &= 2 \{1 - P(|x-\theta|) \}, \end{align} \] where \(P\) is the cumulative distribution function for standard Normal distribution. The above derivation works for \(n=1\). When \(n>1\), the sufficient statistics (sample mean) and the sample variance can be used.

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

contour_normal <- function(x, theta){
  x_bar <- mean(x)
  n <- length(x)
  return(2 * (pnorm(abs(x_bar-theta)*sqrt(n), lower.tail = F) ))
}

Example demonstration

Now we assume the true parameter is \(0\) and sample size is \(1\), let’s plot the contour function given that we observed \(X=1\).

n <- 1
true_theta <- 0
X <- 1

theta <- seq(true_theta-6, true_theta+6, .1)
pi_normal <- contour_normal(X, theta)
plot(theta, pi_normal, type = "l", xlab = expression(theta), ylab = expression(pi[x](theta)))
pi_true_theta <- contour_normal(X, true_theta)
abline(h=.05, lty=2)
segments(-1, 0, 0, 0, col="red", lwd=2)
segments(0, 0, 0, contour_normal(X,0), lty=3, col="red")
points(0, contour_normal(X,0), col="red", pch=19)

As shown above, the black curve is the contour function for \(x=1\). The possibility measure (or upper probability) of the hypothesis \(h=[-1,0]\) can be found by maximization:

\[ \Pi_x(h) = \sup_{\theta \in [-1,0]} \pi_x(\theta) \approx 0.317,\] where the red circle is.

To find a \(95\%\) confidence set, the black dashed line at \(0.05\) shows that the set is approximately \([-1, 3]\).

Normal Distribution

Shih-Ni Prim

2026-08-30

IM construction

Relative likelihood

Contour function

Example demonstration

Reference