Abstract: Methods, apparatus, systems, and articles of manufacture to perform computer-based monitoring of audiences of network-based media using information theory to estimate intermediate level unions are disclosed. An example apparatus to determine a deduplicated, census-based audience metric of media includes panel union calculator circuitry to calculate a threshold statistic corresponding to an intermediate union of a panel hierarchy, and census union calculator circuitry to calculate a deduplicated audience value corresponding to the intermediate union of the census hierarchy based on the threshold statistic.

Abstract: Methods, apparatus, systems, and articles of manufacture are disclosed to estimate population reach from marginals. Example apparatus disclosed herein are to determine an audience relationship value based on first audience exposure values for a recorded audience of a media and a first de-duplicated audience value for the recorded audience of the media, determine bounds of a pseudo universe estimate of a population audience of the media based on the audience relationship value and second audience exposure values for the population audience of the media, and iteratively determine the pseudo universe estimate of the population audience and a second de-duplicated audience value for the population audience of the media based on the audience relationship value and the second audience exposure values, the pseudo universe estimate of the population audience to be within the bounds.

Abstract: Methods and apparatus to estimate cardinality of users represented across multiple bloom filter arrays are disclosed. Examples includes processor circuitry to execute and/or instantiate instructions to generate a first composite Bloom filter array based on first and second Bloom filter arrays. The processor circuitry is to generate a final composite Bloom filter array based on the first composite Bloom filter array and a third Bloom filter array. Different ones of the first, second, and third Bloom filter arrays representative of different sets of users who accessed media. The first, second, and third Bloom filter arrays including differential privacy noise. The processor circuitry to estimate a cardinality of a union of the first, second, and third Bloom filter arrays based on the final composite Bloom filter array.

Abstract: Methods, apparatus, systems and articles of manufacture to estimate and deduplicate audiences are disclosed herein. An example apparatus includes a controller to determine a subunion of at least first and second marginal audiences of media based on of panel data and census data, the panel data including a panel impression count and a panel audience size, and the census data including a census impression count, an audience size calculator to determine a census audience size of the at least the first and second marginal audiences based on the panel impression count and the panel audience size and determine a subunion census audience size, the subunion census audience size corresponding to an overlap between the at least the first and second marginal audiences; and a report generator to generate a report including the census audience size and the subunion census audience size.

Abstract: Methods and systems to estimate audience measurement metrics based on users represented in Bloom filter arrays are disclosed. An apparatus includes a communications interface to receive a first Bloom filter array from a first computer of a first database proprietor. The first Bloom filter array is representative of first users who accessed media. The first users are registered with the first database proprietor. The first Bloom filter array includes a first array of first elements. Values of respective ones of the first elements are either a 0 or a 1 based on whether quantities of the first users allocated to the respective ones of the first elements are even or odd. The apparatus further includes a Bloom filter array analyzer to estimate a first cardinality for the first Bloom filter array. The first cardinality is indicative of a total number of the first users who accessed the media.

Abstract: Methods, apparatus, systems, and articles of manufacture to perform computer-based monitoring of audiences of network-based media using information theory to estimate intermediate level unions are disclosed. An example apparatus to determine a deduplicated, census-based audience metric of media includes panel union calculator circuitry to calculate a threshold statistic corresponding to an intermediate union of a panel hierarchy, and census union calculator circuitry to calculate a deduplicated audience value corresponding to the intermediate union of the census hierarchy based on the threshold statistic.

Abstract: Methods and apparatus to estimate census level audience sizes, impression counts, and duration data are disclosed. Example instructions cause one or more processor to at least set up a system of equations based on first census data; select a census parameter value based on a constraint for census parameter selection, the constraint based on media access represented in panel data; iteratively solve for second census data that satisfies the system of equations using the panel data and the census parameter value, the second census data including an audience size value, an impression count value, and a duration value for a media item accessed by anonymous audience members in a demographic group; and generate a report including the second census data.

Abstract: Methods, apparatus, systems, and articles of manufacture to estimate audience measurement metrics based on users represented in Bloom filter arrays are disclosed. An apparatus includes a communications interface to receive a first Bloom filter array from a first computer of a first database proprietor. The first Bloom filter array is representative of first users who accessed media. The first users are registered with the first database proprietor. The first Bloom filter array includes a first array of first elements. Values of respective ones of the first elements are either a 0 or a 1 based on whether quantities of the first users allocated to the respective ones of the first elements are even or odd. The apparatus further includes a Bloom filter array analyzer to estimate a first cardinality for the first Bloom filter array. The first cardinality is indicative of a total number of the first users who accessed the media.

Abstract: Methods and apparatus to estimate cardinality of users represented across multiple bloom filter arrays are disclosed. Examples includes processor circuitry to execute and/or instantiate instructions to generate a first composite Bloom filter array based on first and second Bloom filter arrays. The processor circuitry is to generate a final composite Bloom filter array based on the first composite Bloom filter array and a third Bloom filter array. Different ones of the first, second, and third Bloom filter arrays representative of different sets of users who accessed media. The first, second, and third Bloom filter arrays including differential privacy noise. The processor circuitry to estimate a cardinality of a union of the first, second, and third Bloom filter arrays based on the final composite Bloom filter array.

Abstract: Methods, apparatus, systems, and articles of manufacture to estimate cardinality of users represented in arbitrarily distributed bloom filter arrays are disclosed. A system includes a communication interface to: access a first Bloom filter array representative of first entries in a first database, the first entries allocated to ones of first elements in the first Bloom filter array based on a non-uniform distribution of outputs of a hash function applied to the first entries, and access a second Bloom filter array representative of second entries in a second database. The system also includes machine readable instructions to cause one or more processors to estimate a cardinality of a union of the first and second entries based on the non-uniform distribution of the outputs of the hash function.

Abstract: Disclosed examples determine constraints based on a structure of initial audience totals; for a number of iterations: select values for modified audience totals that satisfy the constraints, the modified audience totals corresponding respectively to the initial audience totals; and determine an output based on the values for the modified audience totals and the initial audience totals; select final values of the modified audience totals resulting from completion of the number of iterations, the final values of the modified audience totals to be logically consistent; and generate a report including the final values of the modified audience totals.

Abstract: Methods, apparatus, systems, and articles of manufacture to estimate cardinality of users represented in arbitrarily distributed bloom filter arrays are disclosed. An apparatus includes a communications interface to obtain a first Bloom filter array generated by a first computer of a first database proprietor and obtain a second Bloom filter array generated by a second computer of a second database proprietor. The apparatus includes a cardinality analyzer to estimate a total number of unique users included in a union of corresponding ones of the first and second users represented in the first and second Bloom filter arrays regardless of whether the first and second users are mapped, based on a first distribution or a second distribution different than the first distribution, to the corresponding first and second Bloom filter arrays.

Abstract: Methods, apparatus, systems and articles of manufacture to estimate and deduplicate audiences are disclosed herein. An example apparatus includes a controller to determine a subunion of at least first and second marginal audiences of media based on of panel data and census data, the panel data including a panel impression count and a panel audience size, and the census data including a census impression count, an audience size calculator to determine a census audience size of the at least the first and second marginal audiences based on the panel impression count and the panel audience size and determine a subunion census audience size, the subunion census audience size corresponding to an overlap between the at least the first and second marginal audiences; and a report generator to generate a report including the census audience size and the subunion census audience size.

Abstract: Methods, apparatus, systems and articles of manufacture to estimate and deduplicate audiences are disclosed herein. An example apparatus includes a controller to determine a subunion of at least first and second marginal audiences of media based on of panel data and census data, the panel data including a panel impression count and a panel audience size, and the census data including a census impression count, an audience size calculator to determine a census audience size of the at least the first and second marginal audiences based on the panel impression count and the panel audience size and determine a subunion census audience size, the subunion census audience size corresponding to an overlap between the at least the first and second marginal audiences; and a report generator to generate a report including the census audience size and the subunion census audience size.

Abstract: A disclosed example apparatus includes a communication interface to: access impression count data corresponding to a plurality of platforms; and access deduplicated total audience size data; an arithmetic logic unit to: generate a total impressions count by aggregating the impression count data corresponding to the plurality of platforms; generate an estimated per-platform impression count by dividing the total impressions count by a number of the platforms, a solver controller to: instruct a numerical solver to utilize the number of the platforms, the deduplicated total audience size data, and the estimated per-platform impression count to estimate the deduplicated audience size of a first platform of the platforms; and store the deduplicated audience size of the first platform in memory.

Abstract: Methods, apparatus, systems, and articles of manufacture to perform computer-based monitoring of audiences of network-based media using information theory to estimate intermediate level unions are disclosed. An example apparatus to determine a deduplicated, census-based audience metric of media includes panel union calculator circuitry to calculate a threshold statistic corresponding to an intermediate union of a panel hierarchy, and census union calculator circuitry to calculate a deduplicated audience value corresponding to the intermediate union of the census hierarchy based on the threshold statistic.

Abstract: Methods, and apparatus are disclosed to estimate a deduplicated audience of a partitioned audience of media presentations. An example apparatus includes interface circuitry, instructions in the apparatus, and processor circuitry to execute the instructions to at least: identify one or more nodes of a graph structure selected for estimation of a deduplicated audience, and estimate a value indicative of the deduplicated audience across a first selected node of the graph structure and a second selected node of the graph structure.

Abstract: Methods, apparatus, systems, and articles of manufacture are disclosed to determine census audience measurements.

Abstract: Methods and apparatus to estimate audience sizes and durations for media accesses are disclosed. An example apparatus at least one memory, instructions, and processor circuitry to execute the instructions to select constraint equations corresponding to one or more events, the one or more events corresponding to a first audience and a second audience, the first audience to include the second audience, and determine, based on the constraint equations and an auxiliary equation, event audience sizes corresponding to the first audience and to the one or more events.

Abstract: Example technical solutions to determine census-level audience metrics across demographics are disclosed. Disclosed example apparatus include a distribution parameter solver to initialize distribution parameter values for a probability of an individual within a demographic being (i) included in a subscriber audience for the demographic and (ii) having a first impression duration, the subscriber audience having a first subscriber audience size. Disclosed example apparatus also include a divergence parameter solver to determine divergence parameter values between (i) the first subscriber audience size and the first impression duration and (ii) a census-level audience size and a second impression duration based on the initialized distribution parameter values.