Parameter Identifiability of Discrete Bayesian Networks with Hidden Variables

AbstractIdentifiability of parameters is an essential property for a statistical model to be useful in most settings. However, establishing parameter identifiability for Bayesian networks with hidden variables remains challenging. In the context of finite state spaces, we give algebraic arguments establishing identifiability of some special models on small directed acyclic graphs (DAGs). We also establish that, for fixed state spaces, generic identifiability of parameters depends only on the Markov equivalence class of the DAG. To illustrate the use of these results, we investigate identifiability for all binary Bayesian networks with up to five variables, one of which is hidden and parental to all observable ones. Surprisingly, some of these models have parameterizations that are generically 4-to-one, and not 2-to-one as label swapping of the hidden states would suggest. This leads to interesting conflict in interpreting causal effects.

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A simple interpretation of undirected edges in essential graphs is wrong

PLoS ONE ◽

10.1371/journal.pone.0249415 ◽

2021 ◽

Vol 16 (4) ◽

pp. e0249415

Author(s):

Erich Kummerfeld

Keyword(s):

Artificial Intelligence ◽

Directed Acyclic Graphs ◽

Equivalence Classes ◽

The Other ◽

Causal Discovery ◽

Simple Interpretation ◽

Acyclic Graphs ◽

Dutch Books ◽

Markov Equivalence ◽

Causal Directionality

Artificial intelligence for causal discovery frequently uses Markov equivalence classes of directed acyclic graphs, graphically represented as essential graphs, as a way of representing uncertainty in causal directionality. There has been confusion regarding how to interpret undirected edges in essential graphs, however. In particular, experts and non-experts both have difficulty quantifying the likelihood of uncertain causal arrows being pointed in one direction or another. A simple interpretation of undirected edges treats them as having equal odds of being oriented in either direction, but I show in this paper that any agent interpreting undirected edges in this simple way can be Dutch booked. In other words, I can construct a set of bets that appears rational for the users of the simple interpretation to accept, but for which in all possible outcomes they lose money. I put forward another interpretation, prove this interpretation leads to a bet-taking strategy that is sufficient to avoid all Dutch books of this kind, and conjecture that this strategy is also necessary for avoiding such Dutch books. Finally, I demonstrate that undirected edges that are more likely to be oriented in one direction than the other are common in graphs with 4 nodes and 3 edges.

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Do the Flexible Employment Arrangements Increase Job Satisfaction and Employee Loyalty? Evidence from Bayesian Networks and Instrumental Variables

10.20944/preprints201607.0007.v1 ◽

2016 ◽

Author(s):

Eleftherios Giovanis

Keyword(s):

Job Satisfaction ◽

Bayesian Networks ◽

Instrumental Variables ◽

Causal Effect ◽

Directed Acyclic Graphs ◽

Employee Relations ◽

Employee Loyalty ◽

Flexible Employment ◽

Acyclic Graphs ◽

The Relationship

This study explores the relationship between job satisfaction, employee loyalty and two types of flexible employment arrangements; teleworking and flexi-time. The analysis relies on data derived by the Workplace Employee Relations Survey (WERS) in 2004 and 2011. A propensity score matching and least squares regressions are applied. Furthermore, Bayesian Networks (BN) and Directed Acyclic Graphs (DAGs) are employed in order to confirm the causality between employment types explored and the outcomes of interest. Finally, an instrumental variables (IV) approach based on the BN framework is proposed and applied in this study. The results support that there is a positive causal effect from these employment arrangements on job satisfaction and employee loyalty.

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Consistency Guarantees for Greedy Permutation-Based Causal Inference Algorithms

Biometrika ◽

10.1093/biomet/asaa104 ◽

2021 ◽

Author(s):

L Solus ◽

Y Wang ◽

C Uhler

Keyword(s):

Graphical Models ◽

Directed Acyclic Graph ◽

Real Data ◽

Directed Acyclic Graphs ◽

Equivalence Classes ◽

Acyclic Graph ◽

Basic Task ◽

Acyclic Graphs ◽

Edge Graph ◽

Markov Equivalence

Abstract Directed acyclic graphical models are widely used to represent complex causal systems. Since the basic task of learning such a model from data is NP-hard, a standard approach is greedy search over the space of directed acyclic graphs or Markov equivalence classes of directed acyclic graphs. As the space of directed acyclic graphs on p nodes and the associated space of Markov equivalence classes are both much larger than the space of permutations, it is desirable to consider permutation-based greedy searches. Here, we provide the first consistency guarantees, both uniform and high-dimensional, of a greedy permutation-based search. This search corresponds to a simplex-like algorithm operating over the edge-graph of a subpolytope of the permutohedron, called a directed acyclic graph associahedron. Every vertex in this polytope is associated with a directed acyclic graph, and hence with a collection of permutations that are consistent with the directed acyclic graph ordering. A walk is performed on the edges of the polytope maximizing the sparsity of the associated directed acyclic graphs. We show via simulated and real data that this permutation search is competitive with current approaches.

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Learning Markov Equivalence Classes of Directed Acyclic Graphs: An Objective Bayes Approach

Bayesian Analysis ◽

10.1214/18-ba1101 ◽

2018 ◽

Vol 13 (4) ◽

pp. 1235-1260 ◽

Cited By ~ 4

Author(s):

Federico Castelletti ◽

Guido Consonni ◽

Marco L. Della Vedova ◽

Stefano Peluso

Keyword(s):

Directed Acyclic Graphs ◽

Equivalence Classes ◽

Objective Bayes ◽

Acyclic Graphs ◽

Markov Equivalence ◽

Bayes Approach

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The Lorax Problem: Introduction to Bayesian Networks

Bayesian Statistics for Beginners ◽

10.1093/oso/9780198841296.003.0019 ◽

2019 ◽

pp. 325-352

Author(s):

Therese M. Donovan ◽

Ruth M. Mickey

Keyword(s):

Bayesian Networks ◽

Joint Probability ◽

Bayes Theorem ◽

Directed Acyclic Graphs ◽

Causal Models ◽

Influence Diagrams ◽

Acyclic Graphs ◽

New Information ◽

Alternative Hypotheses ◽

The Lorax

The “Lorax Problem” introduces Bayesian networks, another set of methods that makes use of Bayes’ Theorem. The ideas are first explained in terms of a small, standard example that explores two alternative hypotheses for why the grass is wet: the sprinkler is on versus it is raining. The chapter describes how to depict causal models graphically with the use of influence diagrams and directed acyclic graphs. Bayes’ Theorem is used to compute conditional probabilities and to update probabilities once new information is obtained or assumed. The software program Netica is introduced. Finally, the chapter provides a second example of Bayesian networks based on The Lorax by Dr. Seuss. The reader will gain a firm understanding of parent nodes (also known as root nodes), child nodes, conditional probability tables (CPTs), and the chain rule for joint probability.

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