Missing data are ubiquitous in many domain such as healthcare. When these data entries are not missing completely at random, the (conditional) independence relations in the observed data may be different from those in the complete data generated by the underlying causal process.Consequently, simply applying existing causal discovery methods to the observed data may lead to wrong conclusions. In this paper, we aim at developing a causal discovery method to recover the underlying causal structure from observed data that are missing under different mechanisms, including missing completely at random (MCAR),missing at random (MAR), and missing not at random (MNAR). With missingness mechanisms represented by missingness graphs (m-graphs),we analyze conditions under which additional correction is needed to derive conditional independence/dependence relations in the complete data. Based on our analysis, we propose Miss-ing Value PC (MVPC), which extends the PC algorithm to incorporate additional corrections.Our proposed MVPC is shown in theory to give asymptotically correct results even on data that are MAR or MNAR. Experimental results on both synthetic data and real healthcare applications illustrate that the proposed algorithm is able to find correct causal relations even in the general case of MNAR.
more »
« less
Recovering Probability Distributions from Missing Data
A probabilistic query may not be estimable from observed data corrupted by missing values if the data are not missing at random (MAR). It is therefore of theoretical interest and practical importance to determine in principle whether a probabilistic query is estimable from missing data or not when the data are not MAR. We present algorithms that systematically determine whether the joint probability distribution or a target marginal distribution is estimable from observed data with missing values, assuming that the data-generation model is represented as a Bayesian network, known as m-graphs, that not only encodes the dependencies among the variables but also explicitly portrays the mechanisms responsible for the missingness process. The results significantly advance the existing work.
more »
« less
- Award ID(s):
- 1704352
- PAR ID:
- 10060355
- Date Published:
- Journal Name:
- Proceedings of the Ninth Asian Conference on Machine Learning
- Volume:
- PMLR 77
- Page Range / eLocation ID:
- 574-589
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Query-based explanations for missing answers identify which operators of a query are responsible for the failure to return a missing answer of interest. This type of explanations has proven useful, e.g., to debug complex analytical queries. Such queries are frequent in big data systems such as Apache Spark. We present a novel approach to produce query-based explanations. It is the first to support nested data and to consider operators that modify the schema and structure of the data (e.g., nesting, projections) as potential causes of missing answers. To efficiently compute explanations, we propose a heuristic algorithm that applies two novel techniques: (i) reasoning about multiple schema alternatives for a query and (ii) re-validating at each step whether an intermediate result can contribute to the missing answer. Using an implementation on Spark, we demonstrate that our approach is the first to scale to large datasets while often finding explanations that existing techniques fail to identify.more » « less
-
In probabilistic databases the data is uncertain and is modeled by a probability distribution. The central problem in probabilistic databases is query evaluation, which requires performing not only traditional data processing such as joins, projections, unions, but also probabilistic inference in order to compute the probability of each item in the answer. At their core, probabilistic databases are a proposal to integrate logic with probability theory. This paper accompanies a talk given as part of the Gems of PODS series, and describes several results in probabilistic databases, explaining their significance in the broader context of model counting, probabilistic inference, and Statistical Relational Models.more » « less
-
Real-time decision making in IoT applications relies upon space-efficient evaluation of queries over streaming data. To model the uncertainty in the classification of data being processed, we consider the model of probabilistic strings --- sequences of discrete probability distributions over a finite set of events, and initiate the study of space complexity of streaming computation for different classes of queries over such probabilistic strings. We first consider the problem of computing the probability that a word, sampled from the distribution defined by the probabilistic string read so far, is accepted by a given deterministic finite automaton. We show that this regular pattern matching problem can be solved using space that is only poly-logarithmic in the string length (and polynomial in the size of the DFA) if we are allowed a multiplicative approximation error. Then we show how to generalize this result to quantitative queries specified by additive cost register automata --- these are automata that map strings to numerical values using finite control and registers that get updated using linear transformations. Finally, we consider the case when updates in such an automaton involve tests, and in particular, when there is a counter variable that can be either incremented or decremented but decrements only apply when the counter value is non-zero. In this case, the desired answer depends on the probability distribution over the set of possible counter values that can range from 0 to n for a string of length n. Under a mild assumption, namely probabilities of the individual events are bounded away from 0 and 1, we show that there is an algorithm that can compute all n entries of this probability distribution vector to within additive 1/poly(n) error using space that is only Õ(n). In establishing these results, we introduce several new technical ideas that may prove useful for designing space-efficient algorithms for other query models over probabilistic strings.more » « less
-
We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an α-approximate cover using Õ(m(n/k)^1/(α–1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is , even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1 + ε)-approximation algorithm with Õ(mn/kε^2) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing a lower bound of query complexity. Our lower-bound results follow by carefully designing two distributions of instances that are hard to distinguish. In particular, our first lower bound involves a probabilistic construction of a certain set system with a minimum set cover of size αk, with the key property that a small number of “almost uniformly distributed” modifications can reduce the minimum set cover size down to k. Thus, these modifications are not detectable unless a large number of queries are asked. We believe that our probabilistic construction technique might find applications to lower bounds for other combinatorial optimization problems.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
