  We introduce a simple model illustrating the role of context in communication
and the challenge posed by uncertainty of knowledge of context. We consider a
variant of distributional communication complexity where Alice gets some
information $x$ and Bob gets $y$, where $(x,y)$ is drawn from a known
distribution, and Bob wishes to compute some function $g(x,y)$ (with high
probability over $(x,y)$). In our variant, Alice does not know $g$, but only
knows some function $f$ which is an approximation of $g$. Thus, the function
being computed forms the context for the communication, and knowing it
imperfectly models (mild) uncertainty in this context.
  A naive solution would be for Alice and Bob to first agree on some common
function $h$ that is close to both $f$ and $g$ and then use a protocol for $h$
to compute $h(x,y)$. We show that any such agreement leads to a large overhead
in communication ruling out such a universal solution.
  In contrast, we show that if $g$ has a one-way communication protocol with
complexity $k$ in the standard setting, then it has a communication protocol
with complexity $O(k \cdot (1+I))$ in the uncertain setting, where $I$ denotes
the mutual information between $x$ and $y$. In the particular case where the
input distribution is a product distribution, the protocol in the uncertain
setting only incurs a constant factor blow-up in communication and error.
  Furthermore, we show that the dependence on the mutual information $I$ is
required. Namely, we construct a class of functions along with a non-product
distribution over $(x,y)$ for which the communication complexity is a single
bit in the standard setting but at least $\Omega(\sqrt{n})$ bits in the
uncertain setting.
