A Web proxy must accurately predict network performance between itself and its servers and clients in order to make good distillation decisions. In this paper, we show that the current approaches to make such predictions -- either assuming the proxy is well-connected to all servers or using past observations -- are insufficient. We propose a new prediction method, estimation with uncertainty, that will play a crucial role in web proxies. This method can also be useful in domains such as distributed prefetching, distributed query planning, and cache replacement algorithms that take into account the cost of refetching evicted objects.