Abstract:
Disclosed is a system for approximating conditional probabilities using an annotated decision tree where predictor values that did not exist in training data for the system are tracked, stored, and referenced to determine if statistical aggregation should be invoked. Further disclosed is a system for storing statistics for deriving a non-leaf probability corresponding to predictor values, and a system for aggregating such statistics to approximate conditional probabilities.

Description:
TECHNICAL FIELD 
     The present invention generally relates to machine learning techniques and probabilistic reasoning under uncertainty. More particularly, the present invention relates to learning decision trees from data and using learned decision trees to approximate conditional probabilities. 
     BACKGROUND OF THE INVENTION 
     Machine learning techniques are a mechanism by which accumulated data can be used for prediction and other analytical purposes. For example, web site browsing data can be used to determine web sites more likely to be viewed by particular types of users. As another example, product purchase data can be employed to determine products a consumer is likely to purchase, based on prior product purchase history and other information. 
     One type of machine learning technique is decision-tree learning. A decision tree is a structure employed to encode a conditional probability distribution of a target attribute given a set of predictor attributes. For example, the set of predictor attributes may correspond to web sites a user has or has not viewed, or products a user has or has not purchased. The target attribute may then correspond to a web site or product that an analyst is examining to determine whether the user is likely to view or purchase, respectively. Once a decision tree has been constructed, it can be navigated by employing a particular target user&#39;s data to determine answers to future viewing or purchasing queries concerning the target user. 
     A decision tree  10  illustrated in Prior Art FIG. 1 was constructed by a decision-tree learning algorithm for the purpose of predicting a person&#39;s salary based on various attributes associated with the person. The learning algorithm constructed the decision tree  10  using a set of training data, where each record in the training data corresponded to a person. The set of known attributes in the training data includes Age, Gender, Job and Salary, where Age is a continuous attribute, Job is a categorical attribute with three states {Engineer, Lawyer, Researcher} and Salary is a categorical (binary) attribute with states {High, Low}. Salary is referred to as the target attribute for the tree because the tree is used to predict Salary. Other attributes employed in building the tree  10  are referred to as predictor attributes. 
     The decision tree  10  in Prior Art FIG. 1 encodes the conditional probability distribution p(Salary|Age,Gender,Job) learned from the training data. In particular, for assignments of the predictor attributes Age, Gender and Job, the decision tree  10  can be traversed from a root node  12  down to a leaf node  18 , a leaf node  20  and/or a leaf node  16 . The leaf nodes  18 ,  20  and  16  store a probability distribution for Salary. 
     In general, a decision tree is traversed by starting at the root node and following child links until a leaf node is reached. Each non-leaf node is annotated with the name of a predictor attribute to be examined, and each out-going child link from that node is annotated with a value or a set of values for the predictor attribute. Every value of the predictor attribute corresponds to one out-going child link. When the traversal reaches a non-leaf node, the known value of the corresponding predictor attribute is examined, and the appropriate (unique) child link is followed. Non-leaf nodes are referred to as split nodes (or simply splits) in the decision tree. Each split node is annotated with the name of a predictor attribute X, and the node is thus referred to as a split on X. Splits have at least two children. Prior Art FIG. 1 illustrates splits with two children, which create binary trees. It is to be appreciated by one skilled in the art, that although the application describes binary trees, the more general case of non-binary trees can be employed in accordance with the present invention. 
     To illustrate how to traverse and extract conditional probabilities from a decision tree, consider again the tree  10  in Prior Art FIG.  1 . Assume that an analyst desires to predict the salary of a twenty eight year old female engineer. The analyst desires to use the tree  10  to determine p(Salary|Age=28, Gender=female, Job=Engineer). The traversal starts at the root node  12 , which is a split on Age. Consequently, the known value of twenty eight for Age is examined and compared to the values on the out-going edges of the root node  12 . Because twenty eight is less than thirty, the left child edge is traversed and the traversal moves to a node  14 . The node  14  is a split on Job, and because Job=Engineer for the person in question, the traversal moves next to the node  18 . The node  18  is a leaf node, and consequently the traversal completes and the conditional probability for Salary can be obtained. In particular, 
     
       
           P (Salary=High|Age=28, Gender=female, Job=Engineer)=0.65 
       
     
     
       
           P (Salary=Low|Age=28, Gender=female, Job=Engineer)=0.35 
       
     
     Note that the decision tree  10  does not contain any splits on Gender. This means that the learning algorithm identified that Gender was not useful for predicting Salary, at least in the context of knowing Age and Job. 
     In general, given a decision tree for a probability distribution p(Y|X 1 , . . . ,X N ) then for values x 1 , . . . x n  the values p(Y|X 1 =x 1 , . . . X N =x n ) can be extracted by performing the traversal algorithm as described above, and using the distributions stored in the leaf nodes. One skilled in the art will appreciate that p(Y|X 1 , . . . X n ) denotes either a discrete probability distribution or a probability density function, depending on whether Y is a discrete or a continuous attribute, respectively. 
     There are three problems, using decision trees as described. 
     A first problem arises because decision trees are constructed using a finite set of data that may not contain very many examples corresponding to a probability later requested from the decision tree. Since the probability distributions at the leaf nodes are estimated from the training data, conventionally it is possible to extract a probability that may not have a reliable estimate due to this “inadequate training data” problem. 
     Another problem arises when the requested query does not contain a predictor value that may conventionally be employed to traverse a decision tree and thus retrieve a stored probability. This problem typically occurs when not all of the predictor values (e.g., the values of the attributes that define the splits in the decision tree) are provided in a query, yet a conditional probability of the target attribute is still sought. This problem can arise because the conditional probability distribution p(W|X,Y) does not provide adequate information about the probability distribution p(W|X). That is, if the values for one or more predictors are not known, a conventional decision tree may not extract the desired probability. This is known as the missing predictor problem. 
     A third problem arises because the domain (e.g. the set of possible values) for predictor attributes may not be known when the decision tree is constructed, and these domains may have to be estimated from data. For example, if a decision tree is constructed for p(W|X,Y) using a set of training data, and in that data the attribute X appears in one of two distinct states, the training algorithm is likely to assume that X is a binary attribute. This is problematic if X has more than two states, and the tree is later used to extract p(W|X,Y) for the third value of X. This is known as the “new value” problem. 
     These three problems can be illustrated in Prior Art FIG.  1 . To illustrate the inadequate training data problem, assume that the training data contained no data wherein a lawyer was under thirty. In this case, assume that the split on Job in node  14  was chosen by the learning algorithm because it separates the engineers from the researchers, and this distinction is useful when predicting Salary. By the definition of a split node, Lawyer has to correspond to an out-going edge and the learning algorithm chose to group Lawyers with Researchers. If the tree  10  is used to extract query  22 , the probability distribution will be based on Researchers alone, and may not be an accurate distribution for Lawyers. 
     As another example of the inadequate training data problem, suppose that probabilities are not considered accurate unless at least k records matching the query existed in the training data. Using a conventional decision tree, there is no confidence in the accuracy of the returned probability according to the desired constraint. 
     The missing predictor problem is illustrated in FIG. 1 by considering an attempt to extract the probability p(Salary|Job=Engineer). That is, the value for the Age predictor is unknown. A conventional decision tree  10  is unable to provide the desired probability because it is not known to which child of the root node  12  the traversal should follow to reach a leaf node. 
     The new value problem is illustrated in FIG. 1 by considering the query p(Salary|Age&lt;30, Job=Carpenter). Because the learning algorithm assumed that the values of Job were {Engineer, Lawyer, Researcher} when the tree was built, a conventional decision tree cannot be traversed using the given query because there is no out-going edge from node  14  corresponding to Carpenter, and consequently no conditional probability can be returned. 
     In light of the above problems associated with decision trees, the inadequate training data, the missing predictor, and the new value problems, the usefulness of conventional decision trees are limited. Thus, there is a need for a system and method to build and analyze decision trees so that the problems described above are mitigated. 
     SUMMARY OF THE INVENTION 
     The following presents a simplified summary of the invention in order to provide a basic understanding of some aspects of the invention. This summary is not an extensive overview of the invention. It is not intended to identify key or critical elements of the invention or to delineate the scope of the invention. Its sole purpose is to present some concepts of the invention in a simplified form as a prelude to the more detailed description that is presented later. 
     The present invention provides a system and method for using a decision tree to approximate a conditional probability in either (1) the inadequate training data, (2) the missing predictor, or (3) the new value problem situations. The invention concerns both learning decision trees from data and using the resulting trees to answer queries. 
     When a decision-tree learning algorithm constructs a tree from training data, it uses counts from the data, known as sufficient statistics, to calculate probabilities within leaf nodes. Conventionally, these statistics are discarded, while the present invention stores such sufficient statistics explicitly in the leaf nodes to facilitate deriving conditional probabilities for problematic queries. The present invention recognizes when such problematic queries occur, aggregates the sufficient statistics contained within a subset of the leaf nodes below a problematic split node, and uses the aggregated statistics to derive an appropriate approximate probability. 
     The present invention may include a data structure wherein statistics used to generate stored probabilities are not discarded, and are made available to an aggregation algorithm that approximates the probabilities. The aggregating algorithm may utilize the stored statistics to approximate a probability distribution in either of the three problem situations described above. Since such aggregating techniques may not be required for all queries, a program predicting conditional probabilities by analyzing a decision tree may include a component for detecting when aggregation should occur. Further, a component for determining which of the inadequate training, missing predictor, and/or new value problem situations has triggered the need to aggregate may be included. Different aggregation algorithms may be applied, based, at least in part, on the determination of which problem triggered the need to aggregate. Thus, the problems concerning the three situations described above are mitigated and the usefulness of decision trees in computing conditional probabilities is improved. 
     The invention implements an aggregation method operable to approximate queries in problematic situations. The aggregation method collects a set of sufficient statistics for nodes below a problematic internal node a decision tree to facilitate approximating a desired probability. In one example aspect of the present invention, when a split node is encountered during a query-driven traversal of the tree that triggers at least one of the inadequate training data, missing predictor and/or new value problems, the sufficient statistics for all nodes below the triggering split node can be aggregated, facilitating producing a desired, yet conventionally unproducable probability. This aggregation technique can be referred to as the “simple aggregation” method. 
     It is to be appreciated by one skilled in the art that the sufficient statistics collected by such an aggregation method are identical to the sufficient statistics that would correspond to the problematic split node if the decision tree learning algorithm had stopped partitioning the data at the triggering split node, in which case the triggering split node would have been a leaf node. The “simple aggregation” method can be enhanced by caching aggregate statistics corresponding to (internal) split nodes in the split nodes themselves, which facilitates retrieving such aggregation statistics. With the aggregation statistics cached, probabilities can be pre-computed and stored in the (internal) split nodes, eliminating the need to re-derive probabilities, facilitating efficiently retrieving probabilities. 
     An alternative aspect of the present invention provides the “consistent look-ahead aggregation” method to restrict the sufficient statistics that are included in the aggregation to those statistics that are consistent with a given query. For example, while a query may be missing a predictor value at a trigger node, rather than aggregating all the sufficient statistics for nodes below the trigger node, only nodes consistent with the known conditions in the query may be included in the harvest of sufficient statistics that are aggregated to produce the desired probability. For example, referring to FIG.  1  and Query  24 , the sufficient statistics from node  20  would not be included in the aggregation triggered at the root node  12  because the query specifies that Job=Engineer. 
     Although two aggregating methods, the simple and the consistent look-ahead methods, are described herein, it is to be appreciated by one skilled in the art that the present invention is not intended to be limited to these two aggregation methods, and that other aggregation methods may be employed in connection with the present invention. 
     To the accomplishment of the foregoing and related ends, certain illustrative aspects of the invention are described herein in connection with the following description and the annexed drawings. These aspects are indicative, however, of but a few of the various ways in which the principles of the invention may be employed and the present invention is intended to include all such aspects and their equivalents. Other advantages and novel features of the invention may become apparent from the following detailed description of the invention when considered in conjunction with the drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a prior art tree diagram illustrating a conventional decision tree; 
     FIG. 2 is a prior art tree diagram illustrating a conventional decision tree; 
     FIG. 3 is a schematic block diagram illustrating a system for extracting predictions from a decision tree in accordance with an aspect of the present invention; 
     FIG. 4A is a tree diagram illustrating an annotated decision tree in accordance with an aspect of the present invention; 
     FIG. 4B is a tree diagram illustrating an annotated decision tree in accordance with an aspect of the present invention; 
     FIG. 5 is a tree diagram illustrating an annotated decision tree and the computation of a conditional probability in a missing predictor situation in accordance with an aspect of the present invention; 
     FIG. 6 is a tree diagram illustrating an annotated decision tree and the computation of a conditional probability in a missing predictor situation in accordance with an aspect of the present invention; 
     FIG. 7 is a tree diagram illustrating an annotated decision tree and the computation of a conditional probability in a missing predictor situation utilizing the consistent look-ahead aggregation method in accordance with an aspect of the present invention; 
     FIG. 8 is a tree diagram illustrating an annotated decision tree and the computation of a conditional probability in both a inadequate training data and a missing predictor situation utilizing the look-ahead method in accordance with an aspect of the present invention; 
     FIG. 9 is a tree diagram illustrating an annotated decision tree and the computation of a conditional probability in an inadequate training data situation without utilizing a look-ahead method in accordance with an aspect of the present invention; 
     FIG. 10 is a flow chart illustrating a method for extracting predictions from a decision tree in accordance with an aspect of the present invention; 
     FIG. 11 is a segment of pseudocode for a recursive algorithm for retrieving the statistics sufficient to perform the aggregation method; and 
     FIG. 12 is a schematic block diagram of an exemplary operating environment for a system configured in accordance with the present invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention is now described with reference to the drawings, wherein like reference numerals are used to refer to like elements throughout. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It may be evident, however, to one skilled in the art that the present invention may be practiced without these specific details. In other instances, well-known structures and devices are shown in block diagram form in order to facilitate description of the present invention. 
     As used in this application the phrase “relevant training cases” for a split node S (RTC(S)) denotes the set of instances in the training data with values consistent with the split values on the path from the root to the split node S. For example, in Prior Art FIG. 1, RTC(node  14 ) denotes the set of training cases for which Age&lt;30. Note that the relevant training cases for a root node is the set of all training cases. 
     As used in this application, the term “component” is intended to refer to a computer-related entity, either hardware, a combination of hardware and software, software, or software in execution. For example, a component may be, but is not limited to being, a process running on a processor, a processor, an object, an executable, a thread of execution, a program and a computer. By way of illustration, both an application running on a server and the server can be components. 
     FIG. 3 is a schematic block diagram illustrating a system  30  for extracting predictions from a decision tree  32 . The decision tree  32  is constructed by a learning algorithm  34  that examines a set of training data  36 , which is a subset of the possible data  38 . It is to be understood by one skilled in the art that any of the well-known learning algorithms for constructing decision trees may be utilized to construct the decision tree  32 . 
     The system  30  extracts predictions from the decision tree  32  and may include a tracking component  40 . The tracking component  40  monitors data from the training data  36  as it is transmitted to the learning algorithm  34 , and may keep a list of splits made in constructing the decision tree  32 . In addition, the tracking component  40  may keep a list, for each split, of predictor values not encountered at least a certain number of times within the relevant training cases for that split. Thus, the tracking component  40  compiles information that mitigates the inadequate data problem described above. The list of predictor values not encountered at least a certain number of times in the relevant training cases for a split may be stored in the split nodes, as will be described in FIG.  4 . The tracking component may contain a threshold value k, which may be zero, for each predictor value, such that the list of predictor values will contain values that do no occur in at least k cases. If the predictor value does not occur in at least k cases, it is treated as though it did not occur in the data for the split. 
     The system  30  may also include a statistics component  42  that monitors the learning algorithm  34  processes that determine the probabilities for the leaf nodes in the decision tree  32 . When the learning algorithm  34  completes the decision tree  32 , the statistics component  42  may store in the leaf nodes of the decision tree  32  the information and statistics used in generating the probabilities stored in the leaf nodes. For example, for a binary target attribute (e.g. Salary), the learning algorithm may count the number of occurrences of both states of that attribute. For example, in FIG. 1, component  42  would store, for each leaf the corresponding number of cases where Salary=high and Salary=low). It is to be understood by one skilled in the art that different types of leaf distributions for the target attribute (e.g. binomial distributions for binary attributes, Gaussian distributions for continuous attributes) may have other statistics for determining the probabilities stored in the leaf nodes. 
     An aggregating component  46  may also be included in the system  30  to generate approximate probabilities for conditional probability queries. For example, as noted in the discussion of prior art FIG. 1, a possible query for the decision tree  10  (FIG. 1) may be to predict the conditional probability of a high salary given Age&lt;thirty and Job=lawyer. But, as noted above, if there were no lawyers under the age of thirty in RTC(node  14 ), then the resulting conditional probability may be inaccurate and/or irrelevant. The aggregating component  46  may collect statistics and perform calculations to produce an approximation (examples of which are discussed below) to mitigate the inadequate data problem as described above, as well as the other two problems discussed. The aggregating component  46  may access the statistics generated by the statistics component  42  when performing the aggregating function. Representative examples of the operation of the aggregating component  46  will be described in more detail below. 
     A detecting component  48  may be included to determine when the aggregating component  46  should perform its aggregating function and when the requested conditional probability may be retrieved from a leaf in the decision tree  32 . The detecting component  48  thus may determine that any combination of the (1) missing predictor, (2) inadequate training data, and (3) new value problems described above are present for a query presented to the decision tree  32 . For example, one method the detecting component  48  may employ in detecting the inadequate data problem may be to examine the information generated by the tracking component  40  that was tracking predictor values for split nodes in the decision tree  32  that did not occur in the relevant training data for each of those split nodes. When a query arrives requesting the conditional probability based on a predictor whose value did not appear in the training data, the detecting component  48  may trigger the aggregating functionality of the aggregating component  46 . Examples of such detection and the resulting aggregation are provided below. 
     FIG. 4A is a tree diagram illustrating a decision tree  61  annotated in accordance with an aspect of the present invention. While the decision tree  61  may have similar splits and probabilities as the decision tree  60  illustrated in FIG. 2, the decision tree  61  provides additional data stored in each of the nodes  63 ,  65 ,  67 ,  69 , and  71 . For example, statistics used in generating the probabilities stored in the nodes  63 ,  65 ,  67 ,  69 , and  71  may be stored in those nodes respectively. The decision tree  61  may include a root node  63 , upon which a split on attribute A was made. The root node  63  may include a list of predictor values not encountered at least k times for the attribute A in the training data (illustrated as a bracketed list (A 2 ,A 3 ) inside the root node  63 ). The list of predictor values not encountered in the training data at least k times may have been produced by the tracking component  40  (FIG. 3) as the training data  36  was transmitted to the learning algorithm  34 . The list of predictor values not encountered at least k times in the training data may be utilized by the detecting component  48  (FIG. 3) to determine when the aggregating component  46  should produce an approximation. 
     The left child  65  of the root node  63  may indicate that a split on attribute B was made during the learning process. The left child  65  may include a list of predictor values not encountered in RTC(node  65 ) during training (e.g. B 2 ). The right child  67  of the root node  63  may include, in addition to the probability that x will be in a certain state, statistics for determining how the probability was computed. 
     Similarly, the left child  69  and the right child  71  of split node  65  may contain both probabilities and statistics to determine how probabilities were computed. A query  73  may seek a conditional probability for x given A and B. Unlike the conventional decision tree  60  in FIG. 2, the decision tree  61  may produce an approximate probability based on the additional information stored in the leaf nodes  69  and  71 . For example, when the detection component  48  (FIG. 3) determines that the query  73  may not be processed by reading a probability from a leaf node, it may trigger (e.g. set a flag, send a signal, generate an interrupt) the aggregating component  46  (FIG. 3) to perform its aggregating function. The aggregating component  46  (FIG. 3) may then determine the approximate probability that x will be in state x=1 by, for example, performing an aggregating function utilizing stored counts of x=1 and x=2 from multiple leaf nodes in the tree, thereby mitigating the three problems described above. 
     FIG. 4B is a tree diagram illustrating a decision tree  80  annotated in accordance with an aspect of the present invention. While the decision tree  80  may have similar splits and probabilities as the decision tree  60  illustrated in FIG. 2, the decision tree  80  provides additional data stored in each of the nodes  82 ,  84 ,  86 ,  88  and  90 . For example, statistics used in generating the probabilities stored in the nodes  86 ,  88  and  90  may be stored in those nodes respectively. The decision tree  80  may include a root node  82 , which includes a list of predictor values that are not encountered at least a threshold number of times in the training data (illustrated as a bracketed list ( 1 , 3 ) inside the root node  82 ). The list of predictor values not encountered at least a threshold number of times in the training data may have been produced by the tracking component  40  (FIG. 3) as the training data  36  was transmitted to the learning algorithm  34 . The list of predictor values not encountered at least a threshold number of times in the training data may be utilized by the detecting component  48  (FIG. 3) to determine when the aggregating component  46  should produce an approximation. For example, the detecting component  48  (FIG. 3) may determine that an inadequate training data problem exists or that a missing predictor data problem exists by comparing the predictor values provided in a query against the list of sparse predictor values and/or stored splits. The detecting component  48  may determine that a new value problem exists if a provided predictor value matches neither (1) the values corresponding to the child nodes nor (2) the set of predictor values that had no data. Alternatively, the detecting component  48  may keep an external list of the known states for each predictor, and will recognize when a new value is provided by comparing to the list. 
     The left child  84  of the root node  82  may include a list of predictor values not encountered at least a threshold number of times within RTC(node  84 ) during training (e.g. 2). The right child  86  of the root node  82  may include, in addition to the probability that x will be in a certain state, statistics for determining how the probability was computed. For example, in leaf  86 , the probability that x=1 may be computed to be 0.1, determined by dividing 10 by 100, wherein 10 was the number of times x=1 and 100 is the sum of 10+90, the counts of when x=1 and x=2 respectively, as encountered in the training data. It is to be appreciated by one skilled in the art that there are alternative ways of determining probabilities from the counts in this example, and that for probability distributions other than the binomial distribution, other statistics may be employed to compute these probability distributions. 
     Similarly, the left child  88  and the right child  90  of split node  84  may contain both probabilities and statistics to determine how probabilities were computed. A query  92  seeks a similar conditional probability as the query  74  (FIG.  2 ). Unlike the conventional decision tree  60  in FIG. 2, the decision tree  80  may produce an approximate probability based on the additional information stored in the leaf nodes  88  and  90 . For example, when the detection component  48  (FIG. 3) determines that the query  92  may not be processed by reading a probability from a leaf node, it may trigger (e.g. set a flag, send a signal, generate an interrupt) the aggregating component  46  (FIG. 3) to perform its aggregating function. The aggregating component  46  may then determine the approximate probability that x will be in state x=1 by, for example, summing occurrences of x=1 for children of the split node  84  and dividing that sum by the sum of all occurrences of x. Similarly the conditional probability that x will be in state x=2 given predictor A=2 and predictor B=2 may be approximated as illustrated by query and computation  94 , thereby mitigating the missing predictor problem described above. It is to be appreciated by those skilled in the art that such summing and division is but one example of the functionality of the aggregating component  46  and that other aggregation techniques may be utilized for other probability distributions and trees of different orders. 
     FIG. 5 is a tree diagram illustrating a decision tree  100  annotated in accordance with an aspect of the present invention and a computation of a conditional probability triggered by an inadequate training data problem. The decision tree  100  may contain a root node  102  wherein a split was made upon predictor A, with predictor values two and four encountered at least a threshold number of times in the training data, and predictor values one and three not encountered at least a threshold number of times in the training data. Thus, the tracking component  40  produced the list ( 1 , 3 ) associated with the root node  102 . Split node  104  may be similarly annotated with a list of predictor values not encountered at least a threshold number of times within RTC(node  104 ) during training ( 2 ) by the tracking component  40 . The leaf nodes  106  and  108  may contain both probabilities and statistics for determining how probabilities stored in the leaf nodes  106  and  108  were computed. The split node  110  similarly contains a list of predictor values not encountered at least a threshold number of times (illustrated as bracketed numbers ( 2 , 3 ) in the node  110 ) within RTC(node  110 ), wherein the leaf nodes  112  and  114  contain probabilities and statistics. When a query  116  is presented to the decision tree  100 , the detecting component  48  determines that the predictor B=2 was not encountered at least a threshold number of times during training by examining the list ( 2 ) in node  104  and thus the aggregating component  46  may perform its aggregating function to produce an approximate conditional probability. One such possible aggregation may involve summing the occurrences of x in a certain state and dividing that sum by the sum of all occurrences of x, as illustrated in the computations associated with the query  116 . Since the predictor value had inadequate training data at split node  104 , nodes reachable from split node  104  (e.g.  108 ,  112 , and  114 ) may be aggregated by the aggregating component  46  to produce the approximation as illustrated in the computations associated with the query  116 . Thus, an approximation for the probability that x is in state 1 may be computed as shown in query  116  and similarly, an approximation for the probability that x is in state 2 may be computed as shown in query  118 , for example, thereby mitigating the inadequate training data problem. 
     FIG. 6 is a tree diagram illustrating a decision tree  130  annotated in accordance with an aspect of the present invention and one exemplary computation of a conditional probability in both a missing predictor and an inadequate data situation utilizing the consistent look-ahead aggregation technique described above. The root node  132  and its immediate descendants  134  and  136  may include a list of predictor values not encountered at least a threshold number of times within RTC(node  134 ) and RTC(node  136 ), respectively, during training by the tracking component  40  (FIG.  3 ). The leaf nodes  138 ,  140 ,  142  and  144  contain probabilities that x will be in a certain state and the statistics for computing the probabilities. For example, a query  146  seeks the conditional probability of x given predictor Z=3 and predictor W=1. Since the predictor value Z=3 was not seen during training, using the consistent look-ahead aggregation technique, both the left and right descendants of the root  132  may be traversed by the aggregating component  46  (FIG. 3) when aggregating. But on the left child  134  of the root node  132 , information may restrict the aggregation to leaf  138 , since the predictor value W=1 was encountered. That is, the leaf node  140 , corresponding to W=3, is inconsistent with the known predictor value from the query. Since similar information may be unavailable for the right child  136  (e.g. no predictor for Y), both leaves  142  and  144  may be aggregated. Thus, an approximate probability for x given Z=3 and W=1 may be generated further mitigating missing predictor problems. 
     FIG. 7 is a tree diagram illustrating a decision tree  160  annotated in accordance with an aspect of the present invention and the computation of a conditional probability in a missing predictor situation utilizing an aggregation technique. The root node  162  and its descendant split nodes  164 ,  166 , and  174  have been annotated similarly to the trees described above. Similarly, the leaf nodes  168 ,  170 ,  172 ,  176  and  178  have been annotated with the statistics for computing the stored probabilities. A query  180  seeks the conditional probability that x is in a certain state given that predictor A=3, predictor C=2 and predictor D=1. At the root node  162  there is a path for A=3, so the right side of root node  162  may be analyzed. But at split node  166 , inadequate instances where C=2 were encountered within RTC(node  166 ) during training and thus both sides of split  166  will be analyzed. Looking ahead at split node  174  reveals that there were occurrences for predictor D=1, and thus, using the consistent look-ahead aggregation method, the left child  176  of split  174  will be analyzed. Thus, the approximation for the conditional probability sought in the query  180  may be computed by aggregating the statistics in leaf nodes  172  and  176 , as illustrated in the computations accompanying queries  180  and  182 . 
     FIG. 8 is a tree diagram illustrating a decision tree  190  annotated in accordance with an aspect of the present invention and the computation of a conditional probability in both an inadequate training data and a missing predictor situation utilizing the consistent look-ahead aggregation technique. The root node  192  and its descendant split nodes  194 ,  196  and  204  have been annotated similarly to the trees described above in accordance with the present invention. Similarly, the leaf nodes  198 ,  200 ,  202 ,  206  and  208  have been annotated with statistics for computing the probabilities stored for the leaf nodes, in accordance with the present invention. A query  210  presents the decision tree with a missing predictor problem and an inadequate training data problem as described above. The query  210  seeks the conditional probability of x given predictor C=2 and predictor D=1, but the query may contain assignments for predictors A and B. Conventionally, a decision tree may not yield an accurate conditional probability in response to this query because no predictor value for A was provided. But the present invention provides aggregation techniques for mitigating these problems. 
     The consistent look-ahead aggregation method handles these problems as follows. By way of illustration, since no predictor value is provided for the split on the node  192 , the split attribute being A, both paths from the root node  192  may be examined. On the left side of the root node  192 , again there is a missing predictor problem, since no predictor value is provided for the split on node  194 , the split attribute being B. Thus both leaf nodes  198  and  200  below node  194  will be included in the aggregation. On the right side of the root node  192 , there is a predictor value provided for the split on the attribute C at node  196 . But the predictor value provided triggers an inadequate training data problem since the provided predictor value is in the list of predictor values for which inadequate training data was provided. Consequently, the leaf node  202  will be included in the aggregation. Because the consistent look-ahead aggregation method is being used, and the value of D is 1, the leaf node  206  is included in the aggregation and the leaf node  208  is excluded from the aggregation. It is to be appreciated by those skilled in the art that the consistent look-ahead aggregation technique is but one possible look-ahead aggregation technique and that other such techniques may be utilized for other trees and other probability distributions. 
     FIG. 9 is a tree diagram illustrating the annotated tree  190  of FIG.  8  and an alternative method for computing a conditional probability in an inadequate training data situation that does not utilize the consistent look-ahead aggregation method, but rather an alternative aggregation method, the simple aggregation method. In FIG. 9, the simple aggregation technique is employed for query  214  and thus the statistics from leaf nodes  198 ,  200 ,  202 ,  204 ,  206  and  208  are aggregated, despite the fact that node  208  is inconsistent with the assignment D=1 from the query, and despite the fact that node  202  is inconsistent with the assignment C=2 from the query, using the method as illustrated in the computations associated with query  214 . Thus FIGS. 8 and 9 illustrate alternative aggregation techniques that mitigate the inadequate training data problem described above. 
     FIG. 10 is a flow chart illustrating a method for extracting predictions from a decision tree. At step  300 , predictor values are tracked to determine which predictor values for split nodes were encountered. Tracking the predictor values facilitates triggering an aggregation technique used to approximate a conditional probability. At step  302 , the predictor values tracked at step  300  are stored to facilitate triggering aggregation. At step  304 , statistics for computing probabilities are generated. It is to be appreciated by one skilled in the art that different statistics may be generated for a plurality of tree orders and aggregation techniques. The statistics may be utilized in the aggregation technique to approximate a conditional probability, thus mitigating the missing predictor, inadequate training data and new value problems described above. At step  306 , the statistics generated at step  304  are stored to facilitate aggregation. At step  308  a query to retrieve a conditional probability is received. At step  310  a determination is made concerning whether the query will produce a missing predictor situation. For example, supplied predictor values may be compared to split values and known missing predictor values. If the determination at step  310  is YES, then at step  312  an approximate conditional probability may be generated utilizing an aggregation technique that accesses the statistics stored at step  306 . If the determination at step  310  is NO, then at step  314  a determination may be made concerning whether the query received at step  308  generates an inadequate training data situation. For example, supplied predictor values may be compared to split values and known missing predictor values. If the determination at step  314  is YES, then at step  312  an approximate conditional probability may be generated utilizing an aggregation technique accessing the statistics stored at step  306 . If the determination at step  314  is NO then at step  316 , a determination is made concerning whether the query received at step  308  generates a new value situation. If the determination at step  314  is YES, then at step  312  an approximate conditional probability may be generated utilizing an aggregation technique accessing the statistics stored at step  306 . If the determination at step  316  is YES, then at step  312  an approximate conditional probability may be generated utilizing an aggregation technique accessing the statistics stored at step  306 . If the determination at step  316  is NO, that at step  3   18  the requested probability may be retrieved from a leaf in a decision tree. 
     FIG. 11 is a segment of pseudocode  400  for one exemplary recursive implementation of the consistent look-ahead aggregation algorithm utilized to retrieve statistics for producing an approximate conditional probability. It is to be understood by one skilled in the art that the algorithm described in FIG. 11 is but one possible method for traversing a decision tree and returning the stored statistics and thus the described method is not intended to limit the present invention. 
     With reference to FIG. 12, an exemplary environment  710  for implementing various aspects of the invention includes a computer  712 , including a processing unit  714 , a system memory  716 , and a system bus  718  that couples various system components including the system memory to the processing unit  714 . The processing unit  714  may be any of various commercially available processors. Dual microprocessors and other multi-processor architectures also can be used as the processing unit  714 . 
     The system bus  718  may be any of several types of bus structure including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of commercially available bus architectures. The computer  712  memory includes read only memory (ROM)  720  and random access memory (RAM)  722 . A basic input/output system (BIOS), containing the basic routines that help to transfer information between elements within the computer  712 , such as during start-up, is stored in ROM  720 . 
     The computer  712  further includes a hard disk drive  724 , a magnetic disk drive  726 , e.g., to read from or write to a removable disk  728 , and an optical disk drive  730 , e.g., for reading a CD-ROM disk  732  or to read from or write to other optical media. The hard disk drive  724 , magnetic disk drive  726 , and optical disk drive  730  are connected to the system bus  718  by a hard disk drive interface  734 , a magnetic disk drive interface  736 , and an optical drive interface  738 , respectively. The drives and their associated computer-readable media provide nonvolatile storage of data, data structures, computer-executable instructions, etc. for the computer  712 , including for the storage of broadcast programming in a suitable digital format. Although the description of computer-readable media above refers to a hard disk, a removable magnetic disk and a CD, it should be appreciated by those skilled in the art that other types of media which are readable by a computer, such as zip drives, magnetic cassettes, flash memory cards, digital video disks, Bernoulli cartridges, and the like, may also be used in the exemplary operating environment, and further that any such media may contain computer-executable instructions for performing the methods of the present invention. 
     A number of program modules may be stored in the drives and RAM  722 , including an operating system  740 , one or more application programs  742 , other program modules  744 , and program data  746 . The operating system  740  can be any of a variety of commercially available operating systems. 
     A user may enter commands and information into the computer  712  through a keyboard  748  and a pointing device, such as a mouse  750 . Other input devices (not shown) may include a microphone, an IR remote control, a joystick, a game pad, a satellite dish, a scanner, or the like. These and other input devices are often connected to the processing unit  714  through a serial port interface  752  that is coupled to the system bus  718 , but may be connected by other interfaces, such as a parallel port, a game port, a universal serial bus (“USB”), an IR interface, etc. A monitor  754  or other type of display device is also connected to the system bus  718  via an interface, such as a video adapter  756 . In addition to the monitor, a computer typically includes other peripheral output devices (not shown), such as speakers, printers etc. 
     The computer  712  may operate in a networked environment using logical connections to one or more remote computers, such as a remote computer(s)  758 . The remote computer(s)  758  may be a workstation, a server computer, a router, a personal computer, microprocessor based entertainment appliance (e.g., a WebTV® client system), a peer device or other common network node, and typically includes many or all of the elements described relative to the computer  712 , although, for purposes of brevity, only a memory storage device  760  is illustrated. The logical connections depicted include a local area network (LAN)  762  and a wide area network (WAN)  764 . Such networking environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet. 
     When used in a LAN networking environment, the computer  712  is connected to the local network  762  through a network interface or adapter  766 . When used in a WAN networking environment, the computer  712  typically includes a modem  768 , or is connected to a communications server on the LAN, or has other means for establishing communications over the WAN  764 , such as the Internet. The modem  768 , which may be internal or external, is connected to the system bus  718  via the serial port interface  752 . In a networked environment, program modules depicted relative to the computer  712 , or portions thereof, may be stored in the remote memory storage device  760 . It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between the computers may be used. 
     What has been described above includes examples of the present invention. It is, of course, not possible to describe every conceivable combination of components or methodologies for purposes of describing the present invention, but one of ordinary skill in the art may recognize that many further combinations and permutations of the present invention are possible. Accordingly, the present invention is intended to embrace all such alterations, modifications and variations that fall within the spirit and scope of the appended claims. Furthermore, to the extent that the term “includes” is used in either the detailed description or the claims, such term is intended to be inclusive in a manner similar to the term “comprising” as that term is interpreted when employed as a transitional word in a claim.