Methods and apparatus for estimating total unique audiences

A computer-implemented method for determining a unique audience exposed to media, the method comprising:

• logging a plurality of impressions corresponding to media accessed at client devices;
• generating a matrix and a vector based on (a) a first count of first demographic impressions corresponding to the media accessed via a first source and logged by a first database proprietor and (b) a second count of second demographic impressions corresponding to the media accessed via a second source and identified by a second database proprietor, the vector representative of a plurality of ratios of constraints to a count of the logged plurality of impressions, the first demographic impressions corresponding to first users of the first database proprietor, the second demographic impressions corresponding to second users of the second database proprietor;
• and
• determining at least one of a first unique audience size exposed to the media via the first source, a second unique audience size exposed to the media via the second source, or a total unique audience size exposed to the media via the first and second sources based on the matrix and the vector.

A computer-implemented method for determining a unique audience exposed to media, the method comprising:

• logging a plurality of impressions corresponding to media accessed at client devices;
• determining a first enumeration table, the first enumeration table including first values based on a system of non-linear equations associated with (a) a first count of first demographic impressions corresponding to the media accessed via a first source and logged by a first database proprietor, (b) a first count of first users exposed to the media, (c) a second count of second demographic impressions corresponding to the media accessed via a second source and logged by a second database proprietor, and (d) a second count of second users exposed to the media, the first demographic impressions corresponding to first users of the first database proprietor, the second demographic impressions corresponding to second users of the second database proprietor;
• determining a second enumeration table including second values based on a second system of non-linear equations associated with the first values determined in the first enumeration table and the plurality of impressions;
• and
• determining at least one of a first unique audience size of the media accessed via the first source, a second unique audience size of the media accessed via the second source, or a total unique audience size of the media accessed via the first and second sources by solving second expressions using the second values in the second enumeration table.