Incompleteness due to missing attribute values (aka "null values") is very common in autonomous web databases, on which user accesses are usually supported through mediators. Traditional query processing techniques that focus on the strict soundness of answer tuples often ignore tuples with critical missing attributes, even if they wind up being relevant to a user query. Ideally we would like the mediator to retrieve such relevant uncertain answers and gauge their relevance by accessing their likelihood of being relevant answers to the query. The autonomous nature of the databases poses several challenges in realizing this idea. Such challenges include the restricted access privileges, limited query patterns and sensitivity of database and network resource consumption in the web environment. We introduce a novel query rewriting and optimization framework that tackles these challenges. Our technique involves reformulating the user query based on approximate functional depende...