Example methods disclosed herein include accessing a query requesting a metric associated with a number of unique members of an audience of media over an aggregate monitoring interval corresponding to a plurality of component monitoring intervals. Disclosed example methods also include determining respective aggregate interval probability distributions modeling likelihoods of respective monitored individuals being exposed to the media during the aggregate monitoring interval, a first one of the aggregate interval probability distributions for a first one of the monitored individuals being determined by combining parameters of respective component interval probability distributions modeling likelihoods of the first one of the monitored individuals being exposed to the media during respective ones of the component monitoring intervals. Disclosed example methods further include evaluating an audience-level probability distribution determined from the aggregate interval probability distributions to determine the metric to transmit to the computing device in response to the query.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. An apparatus to process queries concerning media audiences, the apparatus comprising: a probability distribution determiner to: access a query from a computing device requesting a metric associated with a number of unique members of an audience of media over an aggregate monitoring interval specified in the query, the aggregate monitoring interval corresponding to a plurality of component monitoring intervals; and determine respective aggregate interval probability distributions modeling likelihoods of respective monitored individuals being exposed to the media during the aggregate monitoring interval, a first one of the aggregate interval probability distributions for a first one of the monitored individuals being determined by combining parameters of respective component interval probability distributions modeling likelihoods of the first one of the monitored individuals being exposed to the media during respective ones of the component monitoring intervals corresponding to the aggregate monitoring interval; a probability distribution evaluator to evaluate an audience-level probability distribution determined from the aggregate interval probability distributions to determine the metric; and a data interface to receive the query via a network and transmit a message including the metric to the computing device via the network. 2. The apparatus of claim 1, wherein the component interval probability distributions for the first one of the monitored individuals are beta probability distributions specified by respective first shape parameters and second shape parameters, the first one of the aggregate interval probability distributions is a beta probability distribution specified by a third shape parameter and a fourth shape parameter, and the probability distribution determiner is further to: retrieve the first shape parameters and the second shape parameters of the component interval probability distributions from memory; and combine the first shape parameters and the second shape parameters of the component interval probability distributions to determine the third shape parameter and the fourth parameter of the first one of the aggregate interval probability distributions. 3. The apparatus of claim 2, wherein the probability distribution determiner is to combine the first shape parameters and the second shape parameters of the component interval probability distributions by at least: combining the first shape parameters and the second shape parameters of the component interval probability distributions according to a first expression to determine the third shape parameter of the first one of the aggregate interval probability distributions; and combining the first shape parameters and the second shape parameters of the component interval probability distributions according to a second expression different from the first expression to determine the fourth shape parameter of the first one of the aggregate interval probability distributions. 4. The apparatus of claim 2, wherein the first shape parameter and the second shape parameter of a first one of the component interval probability distributions for the first one of the monitored individuals are different from the first shape parameter and the second shape parameter of a second one of the component interval probability distributions for the first one of the monitored individuals. 5. The apparatus of claim 1, wherein the probability distribution evaluator is to numerically convolve the aggregate interval probability distributions for the respective monitored individuals to determine the audience-level probability distribution. 6. The apparatus of claim 5, wherein the probability distribution evaluator is further to evaluate the audience-level probability distribution by at least: accessing a query value included in the query; and numerically integrating the audience-level probability distribution based on the query value to determine the metric. 7. A method to process queries concerning media audiences, the method comprising: accessing, by executing an instruction with a processor, a query from a computing device requesting a metric associated with a number of unique members of an audience of media over an aggregate monitoring interval specified in the query, the aggregate monitoring interval corresponding to a plurality of component monitoring intervals; determining, by executing an instruction with the processor, respective aggregate interval probability distributions modeling likelihoods of respective monitored individuals being exposed to the media during the aggregate monitoring interval, a first one of the aggregate interval probability distributions for a first one of the monitored individuals being determined by combining parameters of respective component interval probability distributions modeling likelihoods of the first one of the monitored individuals being exposed to the media during respective ones of the component monitoring intervals corresponding to the aggregate monitoring interval; and evaluating, by executing an instruction with the processor, an audience-level probability distribution determined from the aggregate interval probability distributions to determine the metric to transmit to the computing device in response to the query. 8. The method of claim 7, wherein the component interval probability distributions for the first one of the monitored individuals are beta probability distributions specified by respective first shape parameters and second shape parameters, the first one of the aggregate interval probability distributions is a beta probability distribution specified by a third shape parameter and a fourth shape parameter, and further including: retrieving the first shape parameters and the second shape parameters of the component interval probability distributions from memory; and combining the first shape parameters and the second shape parameters of the component interval probability distributions to determine the third shape parameter and the fourth parameter of the first one of the aggregate interval probability distributions. 9. The method of claim 8, wherein the combining of the first shape parameters and the second shape parameters of the component interval probability distributions includes: combining the first shape parameters and the second shape parameters of the component interval probability distributions according to a first expression to determine the third shape parameter of the first one of the aggregate interval probability distributions; and combining the first shape parameters and the second shape parameters of the component interval probability distributions according to a second expression different from the first expression to determine the fourth shape parameter of the first one of the aggregate interval probability distributions. 10. The method of claim 8, wherein the first shape parameter and the second shape parameter of a first one of the component interval probability distributions for the first one of the monitored individuals are different from the first shape parameter and the second shape parameter of a second one of the component interval probability distributions for the first one of the monitored individuals. 11. The method of claim 7, further including numerically convolving the aggregate interval probability distributions for the respective monitored individuals to determine the audience-level probability distribution. 12. The method of claim 11, wherein the evaluating of the audience-level probability distribution includes: accessing a query value included in the query; and numerically integrating the audience-level probability distribution based on the query value to determine the metric. 13. The method of claim 7, wherein the query is received from the computing device via a network; and further including transmitting a message including the metric to the computing device via the network. 14. A tangible computer readable storage medium comprising computer readable instructions which, when executed, cause a processor to at least: access a query from a computing device requesting a metric associated with a number of unique members of an audience of media over an aggregate monitoring interval specified in the query, the aggregate monitoring interval corresponding to a plurality of component monitoring intervals; determine respective aggregate interval probability distributions modeling likelihoods of respective monitored individuals being exposed to the media during the aggregate monitoring interval, a first one of the aggregate interval probability distributions for a first one of the monitored individuals being determined by combining parameters of respective component interval probability distributions modeling likelihoods of the first one of the monitored individuals being exposed to the media during respective ones of the component monitoring intervals corresponding to the aggregate monitoring interval; and evaluate an audience-level probability distribution determined from the aggregate interval probability distributions to determine the metric to transmit to the computing device in response to the query. 15. The storage medium of claim 14, wherein the component interval probability distributions for the first one of the monitored individuals are beta probability distributions specified by respective first shape parameters and second shape parameters, the first one of the aggregate interval probability distributions is a beta probability distribution specified by a third shape parameter and a fourth shape parameter, and the instructions, when executed, further cause the processor to: retrieve the first shape parameters and the second shape parameters of the component interval probability distributions from memory; and combine the first shape parameters and the second shape parameters of the component interval probability distributions to determine the third shape parameter and the fourth parameter of the first one of the aggregate interval probability distributions. 16. The storage medium of claim 15, wherein to combine the first shape parameters and the second shape parameters of the component interval probability distributions, the instructions, when executed, cause the processor to: combine the first shape parameters and the second shape parameters of the component interval probability distributions according to a first expression to determine the third shape parameter of the first one of the aggregate interval probability distributions; and combine the first shape parameters and the second shape parameters of the component interval probability distributions according to a second expression different from the first expression to determine the fourth shape parameter of the first one of the aggregate interval probability distributions. 17. The storage medium of claim 15, wherein the first shape parameter and the second shape parameter of a first one of the component interval probability distributions for the first one of the monitored individuals are different from the first shape parameter and the second shape parameter of a second one of the component interval probability distributions for the first one of the monitored individuals. 18. The storage medium of claim 14, wherein the instructions, when executed, further cause the processor to numerically convolve the aggregate interval probability distributions for the respective monitored individuals to determine the audience-level probability distribution. 19. The storage medium of claim 18, wherein to evaluate of the audience-level probability distribution, the instructions, when executed, cause the processor to: access a query value included in the query; and numerically integrate the audience-level probability distribution based on the query value to determine the metric. 20. The storage medium of claim 14, wherein the query is received from the computing device via a network; and the instructions, when executed, further cause the processor to transmit a message including the metric to the computing device via the network.