probti

Probti is a theoretical construct used in probability theory and statistical learning to quantify how sensitive a probabilistic model’s output distribution is to small perturbations in inputs or parameters. It serves as a robustness-oriented complement to traditional accuracy metrics, focusing on the stability of predictions rather than just average performance.

Definition and interpretation

For a probabilistic model M that maps inputs x from a domain X to an output distribution P_M(.|x), and for a perturbation family where inputs x' lie within a neighborhood B(x, epsilon) allowed by a chosen norm, the probti of M at x with perturbation size epsilon is defined as the maximum distance between output distributions across all perturbed inputs: Probti(M, x, epsilon) = sup_{x' in B(x, epsilon)} D(P_M(.|x'), P_M(.|x)). Here D is a divergence or distance measure such as Kullback–Leibler divergence, total variation distance, or Wasserstein distance. Different choices of D yield variant notions of perturbation sensitivity.

Key properties

Probti is nonnegative and equals zero when the model’s output distribution remains unchanged under the permitted perturbations. It is nondecreasing in epsilon and depends on the chosen divergence, the input distribution, and the model structure. It does not by itself guarantee practical robustness; rather, it provides a bound on how much predictions can change under specified perturbations.

Computation and usage

Exact calculation is often intractable for high-dimensional models, so researchers rely on estimates, relaxations, or bounds, typically via sampling, Lipschitz-based arguments, or surrogate objectives. Probti is used to compare models, validate robustness claims, and guide the design of more stable architectures, particularly in safety-critical or adversarially sensitive applications.

Limitations

Results can be sensitive to the chosen perturbation model and divergence measure. High-dimensional settings may yield loose bounds, and Probti does not distinguish between benign and harmful perturbations without further context.

History and nomenclature

The term probti is a relatively recent addition to the vocabulary of robustness analysis, a portmanteau of probability and perturbation. It has appeared in theoretical discussions as a conceptual tool rather than a universally standardized metric.