Clarify the interpretation of MAGSAC++'s inlier "likelihood" definition

Determine whether, in the MAGSAC++ derivation for robust geometric fitting, the quantity g(r | σ) is intended to represent the posterior probability p(k_i = 1 | r_i, θ) that correspondence x_i is an inlier given residual r_i and scale σ, or whether equating g(r | σ) to that probability is mathematically incorrect; rigorously reconcile this point by formulating a coherent probabilistic model that consistently defines inlier probabilities and associated IRLS weights.

Background

The paper analyzes the derivation of MAGSAC++ and identifies conceptual issues with how inlier probabilities are introduced. In Derivation Step 2, MAGSAC++ informally defines the "likelihood of a point x_i being an inlier" to be equal to g(r(x_i; θ) | σ), where g is derived from a (truncated) chi distribution over residuals, and then uses this quantity in an IRLS scheme.

The authors argue that treating g(r | σ) as a probability of the inlier event is problematic because it conflates a continuous density with a probability, can exceed 1 or be unbounded (e.g., for χ₁), and lacks a coherent underlying mixture model that would justify such an identification. This ambiguity—whether the interpretation or the equation is wrong—must be clarified to establish a sound probabilistic foundation for the estimator and its weights.