Probabilistic mediated schema

This chapter describes some of the formal foundations for data integration with uncertainty. We define probabilistic schema mappings and probabilistic mediated schemas and show how to answer queries in their presence. With these foundations, we show that it is possible to completely automatically bootstrap a pay-as-you-go integration system. 

The architecture of the system is shown in Fig. 4.1. The system contains a number of data sources and a mediated schema (we omit probabilistic mediated schemas from this figure). When the user poses a query Q, which can be either a structured... 

A probabilistic mediated schema will avoid this problem. Consider a probabilistic mediated schema M that includes M3 and M4, each with probability 0.5. For each of them and each source schema, we generate a probabilistic mapping (Sect. 3). For example, the set of probabilistic mappings pM for Si is shown in Fig. 4.5a, b. 

A probabilistic mediated schema consists of a set of mediated schemas, each with a probability indicating the likelihood that the schema correctly describes the domain of the sources. We formally define probabilistic mediated schemas as follows. 

Expressive power A natural question to ask at this point is whether probabilistic mediated schemas provide any added expressive power compared to deterministic ones. Theorem 8 shows that if we consider one-to-many schema mappings, where one source attribute can be mapped to multiple mediated attributes, then any combination of a p-med-schema and p-mappings can be equivalently represented using a deterministic mediated schema with p-mappings, but may not be represented using a p-med-schema with deterministic schema mappings. Note that we can easily extend the definition of query answers to one-to-many mappings, as one mediated attribute can correspond to no more than one source attribute. 

In contrast, Theorem 9 shows that if we restrict our attention to one-to-one mappings, then a probabilistic mediated schema does add expressive power. 

We now show how to create a probabilistic mediated schema M. Given source tables S, ..., Sn, we first construct the multiple schemas Mi,..., Mp in M, and then assign each of them a probability. 

To complete the fully automatic setup of the data integration system, we consider the problem of consolidating a probabilistic mediated schema into a single mediated schema and creating p-mappings to the consolidated schema. We require that the answers to queries over the consolidated schema be equivalent to the ones over the probabilistic mediated schema. 

The main reason to consolidate the probabilistic mediated schema into a single one is that the user expects to see a single schema. In addition, consolidating to a single schema has the advantage of more efficient query answering queries now need to be rewritten and answered based on only one mediated schema. We note that in some contexts, it may be more appropriate to show the application builder a set of mediated schemas and let her select one of them (possibly improving on it later on). 

Even if we introduce probabilistic schema mappings, none of the listed mediated schemas will return ideal answers. For example, using M prohibits returning correct answers for queries that contain both hPhone and oPhone because they are... 

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