  Many neural networks use the tanh activation function, however when given a
probability distribution as input, the problem of computing the output
distribution in neural networks with tanh activation has not yet been
addressed. One important example is the initialization of the echo state
network in reservoir computing, where random initialization of the reservoir
requires time to wash out the initial conditions, thereby wasting precious data
and computational resources. Motivated by this problem, we propose a novel
solution utilizing a moment based approach to propagate uncertainty through an
Echo State Network to reduce the washout time. In this work, we contribute two
new methods to propagate uncertainty through the tanh activation function and
propose the Probabilistic Echo State Network (PESN), a method that is shown to
have better average performance than deterministic Echo State Networks given
the random initialization of reservoir states. Additionally we test single and
multi-step uncertainty propagation of our method on two regression tasks and
show that we are able to recover similar means and variances as computed by
Monte-Carlo simulations.
