Influence functions are a tool to study robustness. They tell us the effect of perturbing one datapoint on the trained parameters. E.g. by taking $x_i \mapsto (1+\epsilon)x_i$.
How can this be used on latent variable models? If a model pairs each datapoint with a latent representation, i.e. $(x_i, z_i)$, how can we capture the effect of perturbing $x_i$, on $z_i$?
And, does the method of influence functions still apply to removing both $x_i, z_i$ altogether?
