This paper studies document ranking under uncertainty. It is tackled in a general situation where the relevance predictions of individual documents have uncertainty, and are dependent between each other. Inspired by the Modern Portfolio Theory, an economic theory dealing with investment in financial markets, we argue that ranking under uncertainty is not just about picking individual relevant documents, but about choosing the right combination of relevant documents. This motivates us to

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... a ranked list of documents on the basis of its expected overall relevance (mean) and its variance; the latter serves as a measure of risk, which was rarely studied for document ranking in the past. Through the analysis of the mean and variance, we show that an optimal rank order is the one that balancing the overall relevance (mean) of the ranked list against its risk level (variance). Based on this principle, we then derive an efficient document ranking algorithm. It generalizes the well-known probability ranking principle (PRP) by considering both the uncertainty of relevance predictions and correlations between retrieved documents. Moreover, the benefit of diversification is mathematically quantified; we show that diversifying documents is an effective way to reduce the risk of document ranking. Experimental results in text retrieval confirm the theoretical insights with improved retrieval performance.