  Understanding the inductive bias of neural networks is critical to explaining
their ability to generalise. Here, for one of the simplest neural networks -- a
single-layer perceptron with n input neurons, one output neuron, and no
threshold bias term -- we prove that upon random initialisation of weights, the
a priori probability $P(t)$ that it represents a Boolean function that
classifies t points in ${0,1}^n$ as 1 has a remarkably simple form: $P(t) =
2^{-n}$ for $0\leq t < 2^n$.
  Since a perceptron can express far fewer Boolean functions with small or
large values of t (low entropy) than with intermediate values of t (high
entropy) there is, on average, a strong intrinsic a-priori bias towards
individual functions with low entropy. Furthermore, within a class of functions
with fixed t, we often observe a further intrinsic bias towards functions of
lower complexity. Finally, we prove that, regardless of the distribution of
inputs, the bias towards low entropy becomes monotonically stronger upon adding
ReLU layers, and empirically show that increasing the variance of the bias term
has a similar effect.
