Staged Event Trees in Categorical Processes

• Staged event trees are rooted probability trees where vertices in the same stage share identical conditional distributions, capturing context-specific independence.
• They represent the sample space through explicit root-to-leaf paths, encoding structural impossibilities directly without using zero probabilities.
• Advanced models integrate scalable learning algorithms, conjugate Dirichlet priors, and causal semantics, making them applicable in health, transport, and service domains.

Staged event trees, usually called staged trees, are rooted probability trees whose internal vertices are partitioned into stages so that situations in the same stage share identical outgoing conditional distributions. Root-to-leaf paths represent atomic outcomes, and path probabilities are products of edge probabilities along those paths. This construction makes the sample space itself explicit, so structurally impossible outcomes are represented by missing edges rather than by zero probabilities in a Cartesian product model. In consequence, staged event trees generalize Bayesian networks for categorical processes with context-specific independence, asymmetric conditional independence, and non-regular sample spaces, while retaining a generative semantics suitable for inference, prediction, and causal analysis (Filigheddu et al., 2023, Leonelli et al., 2022).

1. Event-tree foundation and stage structure

An event tree is a finite rooted directed tree in which each non-leaf vertex is a situation and each outgoing edge represents the next possible event. For a categorical random vector $X=(X_1,\dots,X_p)$ in a fixed order, an $X$-compatible event tree has vertex set

$V=\{v_0\}\cup \bigcup_i X_{[i]},$

where $X_{[i]}=\times_{j\le i} X_j$, and edges connect $x_{[i-1]}$ to $x_{[i]}$ by appending the next variable’s value. Leaves correspond to complete assignments, and the path to a leaf encodes one atomic event in the sample space (Leonelli et al., 2022).

A staged tree augments this event tree with a coloring of internal vertices. Two situations are in the same stage when their florets have identical labeled outgoing distributions. Equivalently, stage membership imposes equality constraints on floret probability vectors, so a single multinomial parameter vector is shared across all situations in that stage. In visual representations, vertices with the same color therefore indicate tied parameters and equal conditional distributions for the next event (Carli et al., 2020).

This path-based representation differs sharply from product-space models. Because the sample space is the set of root-to-leaf paths, it need not be a Cartesian product. Structural constraints are encoded directly by the tree topology: if a transition is impossible in a given context, the corresponding edge is omitted. This makes staged trees natural for asymmetric processes, structural zeros, and structurally missing values, all of which are difficult to encode transparently in symmetric graphical models (Filigheddu et al., 2023, Shenvi et al., 2020).

A further distinction is stratification. In stratified staged trees, all vertices in the same stage lie at the same depth and therefore correspond to the same variable. This preserves a common variable ordering across all paths and is the constraint adopted by the main software implementations for structural learning. Non-stratified trees remain part of the broader chain event graph literature, but stratification is often the practical compromise between expressiveness and tractability (Carli et al., 2020).

2. Probabilistic semantics, likelihood, and model geometry

If stages are denoted by $U_1,\dots,U_K$, and stage $k$ has $m_k$ outgoing categories, then each situation $u\in U_k$ shares a parameter vector

$X$0

The probability of any root-to-leaf path is the product of the stage parameters attached to the edges traversed by that path (Filigheddu et al., 2023).

With data summarized by counts $X$1 for category $X$2 at situation $X$3, the likelihood factorizes by stage:

$X$4

Pooling counts within each stage gives

$X$5

and the stagewise maximum-likelihood estimator is

$X$6

This stagewise factorization is one of the main statistical advantages of staging: parameter sharing both regularizes estimation and makes the local independence assumptions operational in inference (Filigheddu et al., 2023, Leonelli et al., 2022).

Independent Dirichlet priors per stage preserve conjugacy. If

$X$7

then

$X$8

and the marginal likelihood is a product of Dirichlet-multinomial terms over stages. This supports Bayesian model comparison, smoothing under sparse counts, and posterior predictive inference (Filigheddu et al., 2023, Carli et al., 2020).

The number of free parameters is the sum over stages of one less than the number of outgoing categories:

$X$9

For simple staged trees, which satisfy stages = positions in their coalesced representation, the resulting models are balanced and form regular exponential families, so BIC-based model selection is asymptotically valid. More generally, staged tree models are curved exponential families, with natural parameters, sufficient statistics, and cumulant-generating functions determined by the graphical structure itself (Leonelli et al., 2022, Görgen et al., 2020).

A further statistical issue is score equivalence. For causal analyses, statistically equivalent staged trees may encode different graphical hypotheses while representing the same distribution family. A Bayesian Dirichlet score based on path uniformity and mass conservation, termed BDepu, is score-equivalent for staged trees, so statistically equivalent trees receive the same score under that prior construction (Hughes et al., 2022).

3. Structural learning algorithms and scalable subclasses

Structural learning usually proceeds by fixing or learning a variable order, constructing an event tree, and then searching over stage partitions. In the stagedtrees package, the core constructors are full for a saturated model, indep for a stratum-wise independence model, and as_sevt for converting a fitted Bayesian network to a stratified staged tree. Score-based search algorithms include stages_hc, stages_bhc, stages_fbhc, and stages_bhcr, while clustering-based routines include stages_bj, stages_hclust, and stages_kmeans (Carli et al., 2020).

For the subclass of simple staged trees, the first dedicated structure-learning algorithms are the marginal and total algorithms. The marginal algorithm greedily joins vertex colors level by level and then enforces simplicity downstream, whereas the total algorithm greedily joins positions and directly optimizes total BIC under simplicity. When the variable order is unknown, these can be coupled with greedy order search, although exhaustive search over $V=\{v_0\}\cup \bigcup_i X_{[i]},$0 orders is feasible only for small $V=\{v_0\}\cup \bigcup_i X_{[i]},$1 (Leonelli et al., 2022).

Scalability has driven the development of sparse subclasses. A $V=\{v_0\}\cup \bigcup_i X_{[i]},$2-parents staged tree constrains the minimal DAG representation so that each variable has at most $V=\{v_0\}\cup \bigcup_i X_{[i]},$3 parents. The corresponding learning strategy first learns a sparse DAG, converts it to its staged-tree representation, and then applies backward hill-climbing over stage merges. This preserves the parent bound while exploiting asymmetric equalities unavailable in the DAG alone, and it scales efficiently to substantially larger variable sets than unconstrained generic search (Leonelli et al., 2022).

A different scaling strategy is mean-posterior clustering. This method constructs a totally ordered hyperstage a posteriori by ordering situations according to posterior mean transition probabilities, then allows only consecutive merges in that order. On binary trees this produces a quadratically-scaling structural learning algorithm. The same framework motivates binary-tree representations obtained by resize operators, which can enlarge the effective model space by enabling equality constraints over subsets of outcomes that are inaccessible in a single non-binary floret (Strong et al., 2022).

Incomplete data require different machinery. The observed-data likelihood becomes

$V=\{v_0\}\cup \bigcup_i X_{[i]},$4

where each $V=\{v_0\}\cup \bigcup_i X_{[i]},$5 is the set of compatible paths for an incomplete observation. Because this likelihood no longer factorizes by stage, pseudo-likelihoods such as omit (LOm), first-missing (LFM), and stage-average (LSA) have been proposed, together with structural EM variants EM-HC, EM-BHC, and EM-Simple, which alternate imputation-like E-steps over path responsibilities with structure updates in the M-step (Carter et al., 2024).

4. Compact representations, equivalence, and causal semantics

Staged trees are closely linked to chain event graphs (CEGs). Two situations are in the same position when their colored downstream subtrees are identical. A CEG coalesces each position into a single vertex, preserving path probabilities and therefore the full statistical model while often yielding a much more compact graph for inference. In simple staged trees, stages coincide with positions, giving a maximally compact coalescence by construction (Leonelli et al., 2022, Shenvi et al., 2020).

The relationship with Bayesian networks is two-sided. Any Bayesian network over categorical variables with a fixed topological order can be represented by an equivalent staged tree whose coloring is determined by parent configurations. The converse fails in general: staged trees can encode equalities such as

$V=\{v_0\}\cup \bigcup_i X_{[i]},$6

without implying a symmetric independence that a single DAG could represent. To summarize the symmetric part of a staged tree, one can construct its minimal DAG representation, and to summarize non-symmetric dependencies one can further label edges to obtain an asymmetry-labeled DAG (AL-DAG) with labels such as context, partial, local, and total (Leonelli et al., 2022, Varando et al., 2021).

Causal semantics can also be defined directly on staged trees. For an intervention $V=\{v_0\}\cup \bigcup_i X_{[i]},$7, the interventional distribution is obtained by replacing the relevant conditionals along the tree with point masses, yielding

$V=\{v_0\}\cup \bigcup_i X_{[i]},$8

This supports context-specific causal discovery and motivates the Context-Specific Intervention Discrepancy (CID), a pre-metric that compares staged trees by the interventional statements they induce (Leonelli et al., 2021).

For treatment-effect estimation, staged trees provide a transparent encoding of the order $V=\{v_0\}\cup \bigcup_i X_{[i]},$9, where covariates $X_{[i]}=\times_{j\le i} X_j$0 branch first, then treatment $X_{[i]}=\times_{j\le i} X_j$1, then outcome $X_{[i]}=\times_{j\le i} X_j$2. In this representation, the propensity score $X_{[i]}=\times_{j\le i} X_j$3 is read from the treatment floret and the outcome regression $X_{[i]}=\times_{j\le i} X_j$4 from the outcome floret. Standard estimators such as IPW, AIPW, and the g-formula can then be expressed as pathwise sums and products on the tree. Positivity is visible directly: every reachable treatment floret with $X_{[i]}=\times_{j\le i} X_j$5 must contain both treatment edges with positive probability (Varando et al., 30 Sep 2025).

A fully Bayesian extension places priors not only on stage parameters but also on the stage partition itself. Product partition priors encourage parsimony, distance-based priors promote interpretable stage compositions, and split-and-merge MCMC yields posterior samples of staged trees from which posterior ATEs and structure uncertainty can be derived (Cremaschi et al., 5 Nov 2025).

5. Principal application domains

Health-data analysis has been one of the most explicit demonstrations of the methodology. In invasive fungal infection data, the order $X_{[i]}=\times_{j\le i} X_j$6 was used to encode patient trajectories and structural impossibilities such as intubation after $X_{[i]}=\times_{j\le i} X_j$7. The learned staged tree revealed shared and context-specific conditional distributions, including stage equalities for radiological patterns, differing ICU effects across patient groups, and a severe neutropenic ICU risk profile. In the expanded case study, the minimal DAG and dependence subtrees showed that diagnostic time directly influenced death in the staged tree although the Bayesian network omitted that direct effect (Filigheddu et al., 2023).

Staged trees have also been developed as generative classifiers. A staged tree classifier is a $X_{[i]}=\times_{j\le i} X_j$8-compatible staged tree with the class variable at the root, and its decision rule is

$X_{[i]}=\times_{j\le i} X_j$9

The naive staged tree classifier retains the same number of free parameters and the same prediction complexity as naive Bayes while relaxing its symmetric independence assumptions. Across 14 binary-classification datasets, staged tree classifiers were competitive in AUC and balanced accuracy; in the XOR parity experiment with 10 binary features and training size 200, the naive staged tree classifier achieved approximately $x_{[i-1]}$0 accuracy, whereas naive Bayes misclassified more than $x_{[i-1]}$1 of the test data (Carli et al., 2020).

For discrete longitudinal data, staged trees exploit the natural temporal ordering. Two strategies have been proposed: longitudinal staged trees with Markov assumptions encoded by a DAG, and marginal longitudinal staged trees defined on selected margins such as $x_{[i-1]}$2 or $x_{[i-1]}$3. These constructions address the combinatorial growth of repeated-measures trees while preserving stage-based pooling and context-specific structure. The published examples include depression and dentistry studies, where the models were used to study time-varying treatment effects, hygiene trajectories, and caries progression (Carter et al., 2024).

Large-scale service evaluation has supplied a different class of applications. In transport data, staged trees were learned with bootstrap-based validation of variable orderings, stagings, and AL-DAG edge labels. In airline-passenger data, conditioning on dissatisfied departure experience reduced the probability of high overall satisfaction from $x_{[i-1]}$4 to approximately $x_{[i-1]}$5, whereas increasing cabin and meal satisfaction marginals to $x_{[i-1]}$6 increased the probability of high overall satisfaction from $x_{[i-1]}$7 to $x_{[i-1]}$8. In EU rail-traveler data, staged trees integrated heterogeneous regional and national indicators, and the resulting AL-DAG showed that country, rail length, household income, and passenger demand were the main direct parents of satisfaction (Leonelli et al., 2024).

6. Limitations, validation, and theoretical extensions

The main practical limitations are well established. Variable ordering matters because the chosen sequence determines which contexts are expressible and which equality constraints can be represented. Tree size grows rapidly with the number of variables and categories, producing computational and visualization challenges. Too many small stages can overfit sparse data, while overly aggressive merges can suppress genuine heterogeneity. Rare branches are especially unstable without pooling or prior smoothing (Filigheddu et al., 2023, Carter et al., 2024).

Validation strategies have therefore become part of the methodology. Bootstrap resampling has been used to assess robustness of learned variable orderings, stage partitions, and AL-DAG edge labels, and to stabilize what-if sensitivity analyses. In incomplete-data settings, comparisons between pseudo-likelihood and SEM-based learners show that the relative performance of omission, first-missing approximations, and structural EM depends on the missingness mechanism, sample size, and computational budget (Leonelli et al., 2024, Carter et al., 2024).

Several extensions deepen the underlying theory. Balanced staged trees are toric, and a broader class, including SIP-trees and certain hybrids, admits toric structure after a linear change of coordinates. This places large parts of staged-tree model theory within algebraic statistics, where Gröbner bases, determinantal ideals, and monomial parameterizations become available (Görgen et al., 2021). At the same time, the exponential-family analysis shows that staged trees are curved exponential families, with balanced and simple subclasses providing especially regular likelihood geometry (Görgen et al., 2020).

The chain event graph transformation is also theoretically strong: a backward iterative construction maps any staged tree to a CEG without loss of information, and the mapping is bijective in the sense described in the literature. This means that explicit path-based modeling and compact graphical inference are not competing representations but alternative views of the same probability model (Shenvi et al., 2020).

These developments suggest a mature but still expanding framework. Staged event trees now encompass expressive probability-tree semantics, scalable structure learning, compact coalesced graphs, context-specific causal reasoning, and algebraic and exponential-family characterizations. Their distinguishing feature remains the same throughout these lines of work: equality constraints are imposed directly on the unfolding sample space rather than indirectly through symmetric factorization, making staged event trees a distinctive graphical language for asymmetric categorical processes (Leonelli et al., 2022, Leonelli et al., 2021).