Methods for discovery of a Markov boundary from data constitute one of the most important recent developments in pattern recognition and applied data analysis and modeling, primarily because they offer a principled solution to the variable/feature selection problem and give insight about local causal structure. Even though there is always a single Markov boundary of the response variable in faithful distributions, distributions with violations of the intersection property of probability theory may have multiple Markov boundaries. Such distributions are abundant in practical data-analytic applications, and there are several reasons why it is important to discover and extract all Markov boundaries from such data as a critical step of data analysis. The present invention is a novel fast generative method (termed Generalized-iTIE*) that can discover all Markov boundaries from a sample drawn from a distribution. The new method has been tested with simulated data and then applied to discover Markov boundaries in datasets from several application domains including but not limited to: biology, medicine, economics, ecology, image recognition, text processing, and computational biology.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. A computer-implemented method and system for determining and extracting multiple Markov boundaries of the response variable T from a dataset comprising the following steps: 1) finding a “seed”Markov boundary list of variables M of the response variable T and then determining and outputting an equivalency catalogue Θ which lists variable subsets that contain equivalent information (with respect to T) to the subsets of variables in M using the following steps: a) initializing an equivalency catalogue Θ to known information equivalency relations with respect to T or an empty set if no such relations are known; b) initializing a seed Markov boundary M to be a subset of all variables minus the response variable T; c) initializing a priority queue of variables to be examined for inclusion in the seed Markov boundary M from the remaining variables; d) applying a computer-implemented inclusion sub-process for: i. prioritizing variables in the priority queue for inclusion in the seed Markov boundary M; ii. removing non-eligible variables from the priority queue; iii. inserting in the seed Markov boundary M the highest-priority variable(s) in the priority queue and then removing them from the priority queue; e) applying a computer-implemented elimination/analysis sub-process to remove variables from the seed Markov boundary M and determining and recording in the equivalency catalogue Θ which variable subsets contain equivalent information to the subsets of M; f) iterating steps 1.d) and 1.e) according to a computer-implemented interleaving sub-process until a termination criterion is met providing that variables may be re-ranked after each update of the seed Markov boundary M, or the original ranking may be used throughout the method's operation; 2) constructing multiple Markov boundaries of the response variable T by iterating through all variable subsets A of the seed Markov boundary M and substituting them with all variable subsets B that contain equivalent information to A and have been recorded in the equivalency catalogue Θ; and 3) outputting the seed Markov boundary M and all multiple Markov boundaries of the response variable T contained in the equivalency catalogue Θ as identified in step 2). 2. The computer-implemented method and system of claim 1 in which: a) step 1.d) applying a computer-implemented inclusion sub-process that satisfies the following operating characteristics: (i) all variables that are members of all Markov boundaries of the response variable T are eligible for inclusion in the seed Markov boundary list M, and each such variable is assigned a non-zero priority value by the inclusion process prioritization ranking; (ii) variables with zero inclusion priority values are discarded and never considered again; b) step 1.e) applying a computer-implemented elimination/analysis sub-process that satisfies the following operating characteristics: (i) all and only variables that are probabilistically independent of the response variable T given any subset of the seed Markov boundary M, are discarded and never considered again (whether they are inside or outside the seed Markov boundary M); (ii) recording in the equivalency catalogue Θ that subsets of variables A and B contain equivalent information (with respect to T) if the following four conditions hold: i. a subset of variables A is probabilistically independent of the response variable T given some subset B of the seed Markov boundary M; ii. B is probabilistically independent of the response variable T given A; iii. A is not probabilistically independent of T (i.e., it is associated with T); iv. B is not probabilistically independent of T (i.e., it is associated with T); and c) step 1.f) applying a computer-implemented interleaving sub-process that iterates the inclusion and elimination sub-processes any number of times provided that iterating stops when the following criterion is satisfied: at termination no variable outside the seed Markov boundary M is eligible for inclusion and no variable in M can be removed at termination. 3. The computer-implemented method and system of claim 1 in which step 1.d) applies the inclusion sub-process implemented in the following manner: a) using randomized computerized search over values of standard heuristic functions in the field of feature selection employing univariate association, or similarity, or regression coefficients or other equivalent functions; b) using the observed probability of a variable to remain in the seed Markov boundary after conditioning on many subsets of variables; and c) using application-specific structure of the data generating process and/or application-specific distributional characteristics. 4. A computer-implemented method and system for determining and extracting multiple Markov boundaries of the response variable T from a dataset comprising the following steps: 1) finding a seed Markov boundary list of variables M of the response variable T and determining and recording in an equivalency catalogue Θ which variable subsets contain equivalent information (with respect to T) to the subsets of variables in M using the following steps: a) initializing an equivalency catalogue Θ to be empty; b) initializing a seed Markov boundary list M to be empty; c) initializing a priority queue of variables to be examined for inclusion in the seed Markov boundary M from the remaining variables; d) applying the following computer implemented inclusion sub-process: i. sorting in descending order the variables in the priority queue according to their pairwise association with the response variable T; ii. removing from the priority queue variables with zero association with the response variable T; iii. inserting in the seed Markov boundary M the highest-priority variable A in the priority queue and removing it from the priority queue; e) applying the following computer implemented elimination/analysis sub-process: i. if the priority queue is empty, then removing every member A of the seed Markov boundary M that is probabilistically independent of the response variable T given a subset B of the remaining variables in M; ii. if the priority queue is not empty, then if the last variable A that entered the seed Markov boundary M is probabilistically independent from the response variable T given a subset B of the rest of the variables in M, then removing A from M; iii. if a variable A that has been removed from M in the above step 1.e.ii) because A is probabilistically independent of the response variable T given a subset B of M, then recording in the equivalency catalogue Θ that A and B contain equivalent information (with respect to T) if B is probabilistically independent of the response variable T given A; f) applying the following computer implemented interleaving sub-process by iterating steps 1.d) and 1.e) until the priority queue is empty; 2) constructing the list of all Markov boundaries of the response variable T by iterating through all variable subsets A of the seed Markov boundary M and substituting them with all variable subsets B that contain equivalent information to A and have been recorded in the equivalency catalogue Θ; and 3) outputting the seed Markov boundary M and all multiple Markov boundaries of the response variable T contained in the equivalency catalogue Θ as identified in step 2).