Bayesian Networks Representations, Generalized Imputation, and Synthetic Micro-data Satisfying Analytic Constraints
Yves Thibaudeau, William E. Winkler · 2002
This paper shows how Bayesian Networks can be used to create models for discrete data from contingency tables. The advantage is that the models are created relatively automatically using existing software. The models provide representations that approximately preserve the joint relationships of variables and are easy to apply. The models allow imputation for missing data in contingency tables and for the creation of discrete, synthetic microdata satisfying analytic constraints. Introduction Graphical representation of Bayes Nets and other probabilistic relationships date to Lauritzen and Spiegelhalter (1988). They are used extensively in machine learning. For instance, Figure 2 in Getoor et al. (2001) (reprinted below) demonstrates an efficient representation of Census data. 951 parameters are able to represent a potentially large number of cells in a contingency table (7 billion). Bayes Net software will quickly determine dependency relationships such as those of Figure 2 in Getoor et al. (2001). A mathematical representation is PB(A1, ..., An) = Πi≤n PB(Ai | Parents(A i)). (1) If as shown in Figure 2, a given variable depends on only a few other variables (i.e., parents), then representation (1) is very efficient. If there is no missing data, then computation of the probabilities in (1) is exceedingly rapid (see e.g., Friedman 1997). An Alternative to Hot-deck Imputation Given the representation of Figure 2 (alternatively from equation (1)), one can impute as follows: B_IMPUTE: 1. Start with any variable A i having a missing value. If the parents of A i have nonmissing values, impute the missing value of A i according to the conditional probabilities of Figure 2. 2. If a parent has a missing value, proceed to its parents. Proceed through parents until reaching the highest point in the Figure. If at the highest point, impute the missing value according to the conditional probabilities determined by the parents. If there are no parents, impute the probability according to the observed frequency of the variable. 3. Proceed back down the figure filling in all values until the original variables A i is filled in. 4. Proceed through the remaining variables A j having missing data as in steps 1-3. Notes: 1. As with a hot-deck imputation, the above imputation does not deal with non-ignorable nonresponse. 2. Unlike hot-deck imputation, the above imputation procedure preserves joint probabilities. 3. Unlike hot-deck, the above imputation procedure does not rely on matching against representative donors in the set of records. Imputation that Preserves Edit Relationships If the contingency table represents only those records that have no missing values and that satisfy all edits, then no editfailing records will be represented in a figure comparable to Figure 2. All imputations from such a figure will necessarily satisfy all edits. Note: 1. Because of the parsimony of the representation, the BN (Bayes Net) representation will generally only be a within-epsilon representation. In some situations, it may be possible to impute according the B_IMPUTE procedure in such a manner that the imputation does not satisfy edits. Because the within-epsilon representation covers the overwhelming majority of situations from the original contingency table, most imputations should satisfy edits. Testing for Nonignorable Nonresponse It is intuitive that imputation when there are edit restraints will always be nonignorable. Let X1 and X2 be two variables that are associated by an edit. That is, the values assumed by variable X1 are restricted by the values assumed by variable X2. For instance, if X1 represents marital status and X2 represents age, then an edit might be E = {X1 = married, X2 <=15}. If a record R has values for variables X1 and X2 that coincide with E, then the record fails E. If X1 has a blank value (either because it is missing originally or because it is blanked after passing through an edit program), then the value that must be imputed depends on the edit E and the value (<= 15) in the age variable X2. The missingness in variable X1 is dependent on the value (potentially married) that might be in the variable X2. Synthetic Micro-data Satisfying Analytic Constraints If a large database is given by a contingency table, then it can be approximated by a Bayes Net. If the synthetic data are generated according to the probabilities in the BN, then the synthetic micro-data will preserve most of the analytic properties of the original microdata. Alternative Representation of a Contingency Table for a Census Grim et al. (2001) provide a method of representing a contingency table for a census. They represent it as a mixture of multinomial models in which each individual model satisfies a conditional independence assumption. The conditional independence assumption is that the fields are independent given the indicator of being in an individual model. The methods are computationally tractable and provide an alternative to the BN described above.