What does this equation mean?

This kind of benchmark descends from an older and more basic validation move: check whether an explanation changes when it should have to. Adebayo and colleagues showed that several popular saliency methods for image classifiers produce visually similar output whether the underlying model has been trained at all or left at its random initialization, meaning the “explanation” was substantially independent of what training had actually done [ 6 ] . Stated as a minimal necessary condition, if $\theta$ are a trained model’s weights, $\theta_{\mathrm{rand}}$ the same architecture reinitialized at random, and $\phi(f, x)$ the explanation a method produces for input x under model f , then a method worth trusting should satisfy $\phi(f_\theta, x) \neq \phi(f_{\theta_{\mathrm{rand}}}, x) \quad \text{for typical } x$. An explanation that is identical whether or not training happened cannot be reporting anything about what training did; it is a function of the architecture and the input alone, dressed up as a function of the model. This is a floor, not a ceiling — passing it establishes only that a method is not obviously vacuous, and it is the kind of check that a strikingly large share of published interpretability results have never been run against.