Complexity Results and Approximation Strategies for MAP Explanations (1107.0024v1)

Abstract: MAP is the problem of finding a most probable instantiation of a set of variables given evidence. MAP has always been perceived to be significantly harder than the related problems of computing the probability of a variable instantiation Pr, or the problem of computing the most probable explanation (MPE). This paper investigates the complexity of MAP in Bayesian networks. Specifically, we show that MAP is complete for NP^PP and provide further negative complexity results for algorithms based on variable elimination. We also show that MAP remains hard even when MPE and Pr become easy. For example, we show that MAP is NP-complete when the networks are restricted to polytrees, and even then can not be effectively approximated. Given the difficulty of computing MAP exactly, and the difficulty of approximating MAP while providing useful guarantees on the resulting approximation, we investigate best effort approximations. We introduce a generic MAP approximation framework. We provide two instantiations of the framework; one for networks which are amenable to exact inference Pr, and one for networks for which even exact inference is too hard. This allows MAP approximation on networks that are too complex to even exactly solve the easier problems, Pr and MPE. Experimental results indicate that using these approximation algorithms provides much better solutions than standard techniques, and provide accurate MAP estimates in many cases.

• The paper presents a complexity analysis that reveals the MAP decision problem is NPPP-complete even under strict constraints.
• It introduces a hybrid approximation framework using local search methods like stochastic hill climbing and taboo search to overcome intractability.
• Empirical evaluations demonstrate that the proposed methods reliably outperform traditional heuristics on diverse Bayesian network benchmarks.

Overview of Complexity Results and Approximation Strategies for MAP Explanations

The paper "Complexity Results and Approximation Strategies for MAP Explanations" by James D. Park and Adnan Darwiche provides a detailed examination of the complexity inherent in computing Maximum a Posteriori (MAP) configurations within Bayesian networks. The authors deliver an in-depth analysis of the computational barriers involved and suggest approximation methodologies that could serve intractable instances of MAP problems.

Complexity Analysis

One of the pivotal contributions of the paper is establishing that the decision instance of the MAP problem (D-MAP) is NPPP-complete, presenting a significant complexity contrast compared to the NP-complete status of the most probable explanation (MPE) problem. The paper elucidates that, unlike MPE or Pr, computing MAP requires addressing both the combinatorial optimization and the probabilistic counting nature of Bayesian networks, necessitating a combination of optimization and counting tasks. The authors further demonstrate that MAP maintains its NPPP-completeness even within highly constrained network classes, such as polytrees, without evidence, binary variables, and with parameters approaching deterministic values (very close to 0.5).

This complexity is exacerbated in scenarios where networks are manageable via treewidth properties, with MAP demanding stricter conditions on variable elimination orders compared to MPE or Pr. Specifically, the required elimination order for MAP is such that MAP variables must be processed last, often leading to unfavorable width configurations and increased computational expense.

Approximation Framework

Given the established complexity, the authors transition to presenting a structured framework for the approximation of MAP solutions. Recognizing the practical inability to rely on exact algorithms for many real-world and synthetically complex networks, they propose a hybrid approach that can capitalize on efficient inference when feasible, while utilizing heuristic search techniques for optimization. This is operationalized through local search strategies that iteratively refine solutions based on neighbor configurations in the search space.

Two primary local search methods, stochastic hill climbing and taboo search, are explored, integrated with initialization schemes like MPE and maximum likelihood (ML) initializations. When inference is infeasible due to network complexity, the paper advocates for approximate inference techniques like belief propagation, capable of providing useful approximations for MAP score rankings in local search processes.

Empirical Evaluation

Empirical results in the paper demonstrate the superior performance of the proposed approximation techniques compared to traditional methods, especially in terms of delivering reliable MAP estimates across a diverse set of Bayesian network benchmarks, including both synthetic and real-world datasets. The results emphasize that the improvement over static heuristic methods (MPE, ML) is substantive, validating the potential practical benefits of the local search framework.

Implications and Future Work

The findings and methodologies presented in the paper have significant implications for fields leveraging Bayesian networks—for instance, AI-driven diagnostic systems where inferring the most probable underlying configurations in the presence of uncertainty is crucial. As computational complexity remains a cornerstone constraint in practical applications, the advancement of approximation techniques as proposed is paramount.

Future exploration can aim to refine these methodologies further, seeking to reduce the complexity demands where even approximations become cumbersome. This may involve advancing inferential approximations or blending novel heuristic search methods that better balance computational and decision-theoretic implications within the MAP landscape. The exploration of integration with emerging computational paradigms, such as quantum computing, could also offer promising avenues to manage these inherently complex problems.