Feldman-Cousins Confidence Levels - Toy MC Method

In particle physics, the likelihood ratio ordering principle is frequently used to determine confidence regions. This method has statistical properties that are superior to that of other confidence regions. But it often requires intensive computations involving thousands of toy Monte Carlo datasets. The original paper by Feldman and Cousins contains a recipe to perform the toy MC computation. In this note, we explain their recipe in a more algorithmic way, show its connection to 1-CL plots, and apply it to simple Gaussian situations with boundaries.

💡 Research Summary

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The paper revisits the Feldman‑Cousins (F‑C) construction of frequentist confidence intervals based on the likelihood‑ratio ordering principle, a method that avoids the “flip‑flop” problem of classical Neyman intervals and provides a smooth transition between one‑sided limits and two‑sided intervals. While the method is statistically attractive, its practical implementation requires generating thousands of toy Monte‑Carlo (MC) datasets for each hypothesised true parameter value µ, evaluating the test statistic –2 ln R (equivalently Δχ²), and determining the critical Δχ² value that yields the desired coverage. Feldman and Cousins originally gave a recipe for this, but the procedure can appear opaque to practitioners.

The authors translate the original recipe into a clear, step‑by‑step algorithm that can be directly coded into fitting frameworks:

1. Toy data generation – For a chosen true value µ₀, draw a pseudo‑measurement xₜₒʸ from the assumed probability density (in the examples a unit Gaussian with σ = 1).
2. Compute Δχ² – Evaluate χ²(xₜₒʸ, µ₀) and χ²(xₜₒʸ, µ̂) where µ̂ is the maximum‑likelihood estimate respecting any physical boundaries (e.g. µ̂ = max(0, xₜₒʸ) for a non‑negative signal). Δχ² = χ²(µ₀) − χ²(µ̂).
3. Determine the critical Δχ²_c – From the ensemble of toy experiments, find the value Δχ²_c such that a fraction α of toys have Δχ² < Δχ²_c. This implements the integral condition ∫_{x₁}^{x₂} P(x|µ₀) dx = α in a Monte‑Carlo fashion.
4. Construct the confidence interval – For the observed measurement x₀ compute Δχ²(x₀, µ₀) for every µ₀ on a fine grid. The set of µ₀ for which Δχ²(x₀, µ₀) < Δχ²_c forms the confidence interval