Methods and apparatus to estimate population reach from marginal ratings with missing information

A non-transitory computer-readable medium having stored therein instructions that when executed by a computing system cause the computing system to perform a set of acts comprising:

• determining a first ratings-independent universe estimate for a first audience total corresponding to a first union of a first piece of media and a second piece of the media, a first individual audience of the first piece of media to be known and a second individual audience of the second piece of the media to be known;
• determining a second ratings-independent universe estimate for a second audience total corresponding to a second union of a third piece of the media and a fourth piece of the media, the determination of the second ratings-independent universe estimate based on an audience constant, at least one of the third piece of the media or the fourth piece of the media corresponding to an unknown audience portion included in the second audience total of the media;
• and
• determining an estimate of the unknown audience portion based on the second ratings-independent universe estimate for the second audience total.

A method comprising:

• determining, by a computing system, a first ratings-independent universe estimate for a first audience total corresponding to a first union of a first piece of media and a second piece of the media, a first individual audience of the first piece of media to be known and a second individual audience of the second piece of the media to be known;
• determining, by the computing system, a second ratings-independent universe estimate for a second audience total corresponding to a second union of a third piece of the media and a fourth piece of the media, the determination of the second ratings-independent universe estimate based on an audience constant, at least one of the third piece of the media or the fourth piece of the media corresponding to an unknown audience portion included in the second audience total of the media;
• and
• determining, by the computing system, an estimate of the unknown audience portion based on the second ratings-independent universe estimate for the second audience total.