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 possible answers and gauge their relevance by accessing their likelihood of being pertinent answers to the query. The autonomous nature of web databases poses several challenges in realizing this objective. Such challenges include the restricted access privileges imposed on the data, the limited support for query patterns, and the bounded pool of database and network resources in the web environment. We introduce a novel query rewriting and optimization framework QPIAD that tackles these challenges. Our technique involves reformulating the user query based on mined correlations among the database attributes. The reformulated queries are aimed at retrieving the relevant possible answers in addition to the certain answers. QPIAD is able to gauge the relevance of such queries allowing tradeoffs in reducing the costs of database query processing and answer transmission. To support this framework, we develop methods for mining attribute correlations (in terms of Approximate Functional Dependencies), value distributions (in the form of Naïve Bayes Classifiers), and selectivity estimates. We present empirical studies to demonstrate that our approach is able to effectively retrieve relevant possible answers with high precision, high recall, and manageable cost.