1. A computer-implemented method, comprising: accessing, by a data processing apparatus, query data indicating query inputs received from user devices of a plurality of users, the query data also indicating an input context that describes, for each query input, an input context of the query input that is different from content described by the query input and describing a circumstance of the input of the query input at the user device from which the query input was received; grouping, by the data processing apparatus, the query inputs into context clusters based, in part, on the input context for each of the query inputs and the content described by each query input; determining, by the data processing apparatus, for each of the context clusters, a context cluster probability based on respective probabilities of entry of the query inputs that belong to the context cluster, the context cluster probability being indicative of a probability that at least one query input that belongs to the context cluster and provided for an input context of the context cluster will be selected by the user; storing, in a data storage system accessible by the data processing apparatus, data describing the context clusters and the context cluster probabilities; receiving, from a user device, an indication of a user event that includes data indicating a context of the user device, wherein: the user event comprises an indication that the user is accessing a search engine input in a user interface, and the user has not provided to the user interface a query input; and the context is independent of a query input at the user device and describes circumstances of the of the user device at the time of the user event; and accessing the data describing the context clusters and the context cluster probabilities; selecting as a selected context cluster, based on the context cluster probabilities for each of the context clusters and the context of the user device, a context cluster for selection input by the user device; and in response to the selection, providing, to the user device, data that causes the user device to display a context cluster selection input that indicates the selected context cluster for user selection.