Max-linear Bayesian Networks (MLBNs)

• MLBNs are probabilistic graphical models that recursively define variables as the maximum over scaled parent values and innovation terms, capturing extreme events.
• They employ tropical algebra and polyhedral theory to reveal unique conditional independence and identifiability properties that differ from traditional models.
• Applications include risk management and extreme value statistics, using tailored estimation and inference techniques for modeling heavy-tailed data.

Max-linear Bayesian Networks (MLBNs) are a class of probabilistic graphical models in which random variables are recursively defined as the maximum over scaled parent variables and a noise (innovation) term, with the underlying structure encoded by a weighted directed acyclic graph (DAG). MLBNs are fundamentally suited to modeling causal mechanisms and dependence structures in settings dominated by extreme or heavy-tailed events, such as in extreme value statistics, risk management, and networked systems where "winner-takes-all" propagation dominates. Unlike traditional additive or linear Bayesian networks, MLBNs replace addition in the structural equations with the max operator, thereby producing distinct conditional independence and faithfulness properties, unique estimation and inference techniques, and geometric structure best described using tropical algebra and polyhedral theory.

1. Structural Equations and Model Definition

MLBNs are defined by recursive structural equations of the form

$$X_i = \left( \bigvee_{j \in \text{pa}(i)} c_{ji} X_j \right) \vee Z_i,$$

where:

• $\text{pa}(i)$ denotes the set of parent nodes of $i$ in the DAG,
• $c_{ji} > 0$ is the edge weight from $j$ to $i$,
• $Z_i$ are independent, atom-free, positive innovations (e.g., heavy-tailed with Fréchet margins in the extreme value context),
• $\vee$ denotes the maximum operator.

The max-linear coefficient matrix $B$ captures the effective influence from every ancestor $u$ to descendant $v$, with entries

$$b_{vu} = \bigvee_{T \in \text{paths}(u \to v)} \prod_{e \in T} c_e,$$

where the maximum runs over all directed paths $T$ from $u$ to $v$. Thus, each $X_v$ can be represented as a max-linear combination over all ancestors and noise variables:

$X_v = \bigvee_{u \in \text{An}(v)} b_{vu} Z_u.$

Discrete MLBNs, in which $Z_i$ take values in a finite state space, provide an isomorphic extension of conjunctive Bayesian networks in the binary case, with the state space forming a distributive lattice over order-preserving maps from the transitive closure of the DAG to the chain of states (Hollering et al., 2021).

2. Conditional Independence, Faithfulness, and Separation Criteria

The conditional independence (CI) structure in MLBNs departs substantially from classical models. The "winner-takes-all" propagation confers additional CI relations not predicted by d-separation:

• While MLBNs satisfy the global Markov property with respect to their DAG (i.e., d-separation induces CI), many more CIs may arise due to dominance of max-weighted (critical) paths—implicating a lack of faithfulness (Klüppelberg et al., 2019, Améndola et al., 2020).
• Faithfulness fails generically: not all CIs correspond to separations in the underlying DAG, as the largest path can "mask" dependencies indicated by the graph (Klüppelberg et al., 2019).

To address the subtleties in MLBN CI structure, alternative separation criteria have been introduced:

• **-separation* is a refinement suitable for the unweighted case, requiring that paths have at most one collider active, corresponding more closely with the propagation of extremes (Améndola et al., 2021, Améndola et al., 19 Aug 2025).
• $C^\ast$-separation or "critical path separation" is defined relative to the weighted coefficient matrix. Two nodes are $C^\ast$-separated given $L$ if, after removing all paths whose critical weights avoid $L$, no max-connecting path remains (Boege et al., 29 Apr 2025, Améndola et al., 19 Aug 2025). This dependence on both the graph structure and edge weights is fundamental to the MLBN notion of a maxoid (see Section 5).

The Markov equivalence classes of MLBNs coincide with those defined by regular CI from d-separation, meaning that equivalence classes of DAGs under MLBN CI match the classical case (Améndola et al., 2021).

3. Estimation, Inference, and Identifiability

Parameter estimation in MLBNs is distinct due to non-dominated families and non-faithfulness:

• Generalized Maximum Likelihood Estimation (GMLE): For a simple DAG $X_1 \to X_2$ with edge weight $c$, the estimated value is the smallest observed value of the ratio $X_2/X_1$, which has an atom at $c$. For general networks, the GMLE uses observed atoms in ratio distributions to reconstruct coefficients and the minimal max-linear DAG (Klüppelberg et al., 2019).
• Scaling Techniques: In settings where innovations have regularly varying tails, the causal order and ML coefficients can be nonparametrically estimated using empirical spectral measures and scaling relations among the maxima of variable subsets, yielding asymptotic normality for estimated scalings and dependence parameters (Klüppelberg et al., 2019).
• Identifiability: Even with latent (unobserved) variables, parameters are identifiable if the latent nodes satisfy specific criteria, such as having at least two children and acting as sources in the block structure (for trees of transitive tournaments) (Segers et al., 2022). The minimal maximal-weighted path structure (weighted transitive reduction) is uniquely estimable from observational data (Améndola et al., 15 Nov 2024, Klüppelberg et al., 2019).
• Noise and Inference: Incorporation of log-normal or additive noise moves the support of key statistics from discrete atoms to normal mixture components. Estimators remain normally distributed (e.g., the edge parameter estimator for $i \to j$ is $N(\omega_{ij}, \sigma_i^2 + \sigma_j^2)$), and both EM-based Gaussian mixture inference and geometric methods (hyperplane fitting via polytrope boundaries) are valid, with trade-offs depending on the amount and distribution of edge-specific data (Adams et al., 1 May 2025).

4. Geometric and Algebraic Structure: Tropical and Polyhedral Aspects

The algebraic and geometric properties of MLBNs are best framed in terms of tropical semirings and polyhedral theory:

• Tropical Algebra: Under logarithmic coordinates, the max-times semiring underlying MLBN equations becomes max-plus, recasting multiplication as addition and maximum as maximum. The solution $X = C^\ast \odot Z$ involves the tropical Kleene star $C^\ast$, computed as the pointwise maximum over all path weights (Améndola et al., 15 Nov 2024, Boege et al., 29 Apr 2025).
• Polytropes and Statistical Identifiability: The support of the log-transformed variable vector forms a polytrope—a classically convex, tropically convex polytope—described by a system of inequalities $x_i - x_j \leq c_{ij}$ for all $(i,j)$ indexed by $C^\ast$ (Améndola et al., 15 Nov 2024). The facet structure is determined by the weighted transitive reduction. Only the weights on non-redundant (facet-defining) edges are identifiable from extreme event data.
• Maxoids and Polyhedral Fans: The set of all CI relations (the maxoid) induced by a given coefficient matrix via $C^\ast$-separation partitions the space of possible weight matrices into polyhedral cones, which form a full-dimensional fan in weight space. Generic choices of $C$ correspond to unique regions determined by the critical (max-weighted) paths, while the boundaries correspond to degenerate situations (multiple equal-weighted critical paths) (Boege et al., 29 Apr 2025).

5. Conditional Independence and Causal Discovery Algorithms

• Impact Graphs and Source DAGs: In a realized MLBN, impact graphs (random subgraphs) record which parent innovations actually determine maxima at each node, corresponding to "active paths" for each data configuration. Source DAGs are context-specific reductions of the original DAG reflecting remaining variability after partial observation (Améndola et al., 2020).
• Causal Discovery Algorithms: Traditional constraint-based algorithms (e.g., PC algorithm) are inconsistent for MLBNs under d-separation due to violated faithfulness. The adapted PC algorithm retains consistency if the oracle CI test is given by -separation; the resulting CPDAG matches that obtained by d-separation (Améndola et al., 19 Aug 2025). The newly introduced **PCstar algorithm* accepts CI oracles operating under $C^\ast$-separation (i.e., weight-sensitive critical path separation), and additionally orients edges in cycles by exploiting extra (non-minimal) separating sets, achieving higher orientation accuracy in the recovered CPDAG for weighted MLBNs (Améndola et al., 19 Aug 2025).
• Algorithmic Complexity and Implementation: PCstar maintains polynomial complexity in the in-degree ($O(n^{d+2})$), with computational experiments substantiating skeleton and orientation recovery under simulated and realistic MLBNs up to moderate network sizes (Améndola et al., 19 Aug 2025).

6. Applications, Extensions, and Theoretical Directions

MLBNs have been employed in:

• Extreme Value and Risk Modeling: Capturing propagation of catastrophic or rare events in domains such as finance (industry returns), environmental sciences (flooding, weather extremes), and epidemiology. The tail dependence structure of MLBNs is characterized using angular (spectral) measures, supporting analysis of extremal influence and Markov properties in specific network topologies, such as trees of transitive tournaments (Segers et al., 2022, Klüppelberg et al., 2019).
• Algebraic Statistics and Toric Models: Discrete MLBNs generalize conjunctive Bayesian network models and correspond to toric varieties associated with distributive lattices; thus, Gröbner basis methods and toric geometry become directly applicable for understanding model invariants and identifiability (Hollering et al., 2021).
• Open Problems and Ongoing Research: These include development of nonparametric CI tests tailored to MLBNs, further exploration of algebraic/tropical constraints dictating MLBN CIs, integration with interventions and "do-calculus," and extension to models with dependent innovations or noise propagation (Améndola et al., 2020, Améndola et al., 2021, Boege et al., 29 Apr 2025, Améndola et al., 19 Aug 2025).

7. Summary of Fundamental Characteristics

Aspect	MLBNs	Classical BNs	Gaussian SEMs
Structure	Max-linear recursion via DAG, weighted edges	DAG, usually additive	DAG, linear additive
CI mechanism	Winner-takes-all, weight- and path-dependent	d-separation	d-separation
Faithfulness	Generally not faithful (extra CIs arise)	Assumed/typical	Typical
Separation criterion	*-separation and $C^\ast$-separation	d-separation	d-separation
Algebraic/statistical tools	Tropical algebra, maxoids, polytropes	Standard algebra, imsets	Covariance analysis
Identifiability	Up to weighted transitive reduction	DAG and parameters	DAG and parameters
Inference algorithms	PCstar, geometric (polytrope), GMLE/EM	PC, GES, score-based	PC, rank-based CIs

MLBNs occupy a unique domain in the landscape of Bayesian networks, synthesizing extreme value theory, tropical geometry, and algebraic statistics to model systems dominated by extremes and max-propagation. Their distinguishing properties—non-faithfulness to d-separation, nuanced identifiability structure, and geometric underpinnings—necessitate tailored estimation, inference, and causal discovery algorithms, with ongoing advances connecting polyhedral geometry and probabilistic graphical modeling.