We thank all the reviewers for their constructive feedback. Below we provide specific responses to each reviewer. We will add more results in the paper. In the following response 2, we further highlight our important improvements ignored by existing work. The Method In Fig.1(e), Tables 4 and 5, S-GWL can be slightly worse than GWL on node correctness.

Machine learning has been adapted to help solve NP-hard combinatorial optimization problems. One prevalent way is learning to construct solutions by deep neural networks, which has been receiving more and more attention due to the high efficiency and less requirement for expert knowledge. However, many neural construction methods for Vehicle Routing Problems (VRPs) focus on synthetic problem instances with limited scales and specified node distributions, leading to poor performance on real-world problems which usually involve large scales together with complex and unknown node distributions. To make neural VRP solvers more practical in real-world scenarios, we design an auxiliary policy that learns from the local transferable topological features, named local policy, and integrate it with a typical constructive policy (which learns from the global information of VRP instances) to form an ensemble policy. With joint training, the aggregated policies perform cooperatively and complementarily to boost generalization. The experimental results on two well-known benchmarks, TSPLIB and CVRPLIB, of travelling salesman problem and capacitated VRP show that the ensemble policy consistently achieves better generalization than state-of-the-art construction methods and even works well on real-world problems with several thousand nodes.

This paper examines the interdependence generated between two parent nodes with a common instantiated child node, such as two hypotheses sharing common evidence. The relation so generated has been termed "intercausal." It is shown by construction that inter-causal independence is possible for binary distributions at one state of evidence. For such "CICI" distributions, the two measures of inter-causal effect, "multiplicative synergy" and "additive synergy" are equal. The well known "noisy-or" model is an example of such a distribution. This introduces novel semantics for the noisy-or, as a model of the degree of conflict among competing hypotheses of a common observation. In a general Bayesian network, the relation between a pair of nodes can be predictive, meaning we are interested in the effect of a node upon its successors, or, oppositely, diagnostic, where we infer the state of a node from knowledge of its successors.

Monte-Carlo Tree Search (MCTS) methods are drawing great interest after yielding breakthrough results in computer Go. This paper proposes a Bayesian approach to MCTS that is inspired by distributionfree approaches such as UCT [13], yet significantly differs in important respects. The Bayesian framework allows potentially much more accurate (Bayes-optimal) estimation of node values and node uncertainties from a limited number of simulation trials. We further propose propagating inference in the tree via fast analytic Gaussian approximation methods: this can make the overhead of Bayesian inference manageable in domains such as Go, while preserving high accuracy of expected-value estimates. We find substantial empirical outperformance of UCT in an idealized bandit-tree test environment, where we can obtain valuable insights by comparing with known ground truth. Additionally we rigorously prove on-policy and off-policy convergence of the proposed methods.