diff --git a/content/03.overview.md b/content/03.overview.md index 9040a8f..90f06e3 100644 --- a/content/03.overview.md +++ b/content/03.overview.md @@ -67,7 +67,7 @@ Before diving into flexible estimators of $f(c)$, we review early modeling strat #### Conditional and Clustered Models -One approach is to group observations into C contexts, either by manually defining conditions (e.g. male vs. female) or using unsupervised clustering. Each group is then assigned a distinct parameter vector: +One approach is to group observations into C contexts, either by manually defining conditions (e.g. male vs. female) or using unsupervised clustering. The partition can also be learned in a supervised, model-based way: model-based recursive partitioning fits a parametric model and splits the covariate space wherever its coefficients show the strongest instability, recursing to produce interpretable subgroups, with toolkits such as partykit providing standard implementations of these conditional-inference and model-based trees [@doi:10.1198/106186008X319331; @hothorn2015partykit]. Each group is then assigned a distinct parameter vector: $$ \{\widehat{\theta}_0, \ldots, \widehat{\theta}_C\} = \arg\max_{\theta_0, \ldots, \theta_C} \sum_{c \in \mathcal{C}} \ell(X_c; \theta_c), diff --git a/content/06.explicit.md b/content/06.explicit.md index 62e9fb6..dd58b85 100644 --- a/content/06.explicit.md +++ b/content/06.explicit.md @@ -45,7 +45,7 @@ Building on these advances, later research recognized that networks themselves m The methods above pool information across a fixed set of discrete networks, but in many systems the network itself rewires as an underlying context such as time or developmental stage advances. Kolar and Xing formalized this regime as the varying-coefficient varying-structure (VCVS) graphical model, in which both the edge set and the edge weights of a Gaussian graphical model are treated as functions of context rather than fixed quantities [@doi:10.1214/12-EJS739]. When the context is time and the structure changes abruptly, a temporally smoothed $\ell_1$ penalty recovers both the changepoints and the precision matrix on each piecewise-constant block, and these were the first estimators of this kind shown to be sparsistent with established convergence rates [@doi:10.1214/12-EJS739; @kolar2009sparsistent]. When structure instead drifts smoothly, kernel reweighting and total-variation penalties estimate a separate network at each point along the context axis, as in the TESLA method and related time-varying network estimators applied to rewiring gene-regulatory and political networks [@doi:10.1073/pnas.0901910106; @doi:10.1214/09-AOAS308]. The context need not be temporal: TREEGL estimates a tree of networks evolving along a branching biological lineage, borrowing strength between a parent cell type and its descendants while exposing the edges that switch at each division [@doi:10.1093/bioinformatics/btr239]. Pushing the idea further, personalized regression and Bayesian edge-regression models let structure vary at the level of the individual sample, recovering subject-specific networks indexed by clinical covariates or latent similarity rather than by a shared group label [@doi:10.1093/bioinformatics/bty250; @doi:10.1080/01621459.2021.2000866]. The unifying move across these methods is to make the discrete graph structure, and not only the continuous coefficients, an explicit function of context $f(c)$, which is what separates VCVS models from smooth varying-coefficient models that hold the support fixed. **Piecewise-Constant and Partition-Based Models.** -Here, model parameters are allowed to remain constant within specific regions or clusters of the context space, rather than vary smoothly. Approaches include classical grouped estimators and modern partition models, which may learn changepoints using regularization tools like total variation penalties or the fused lasso. This framework is particularly effective for data with abrupt transitions or heterogeneous subgroups. +Here, model parameters are allowed to remain constant within specific regions or clusters of the context space, rather than vary smoothly. Approaches include classical grouped estimators and modern partition models, which may learn changepoints using regularization tools like total variation penalties or the fused lasso. A complementary, tree-structured route learns the partition by recursive splitting: model-based recursive partitioning fits a parametric model, tests its coefficients for instability across candidate context variables, and splits on the variable with the strongest instability before recursing, so that the boundaries are chosen by the data rather than fixed in advance [@doi:10.1198/106186008X319331]. Toolkits such as partykit make these conditional-inference and model-based trees readily usable, fitting a parametric model within each leaf [@hothorn2015partykit]. This framework is particularly effective for data with abrupt transitions or heterogeneous subgroups. A key design principle is that explicit splits of the context space can emulate distinct tasks, clarifying where parameters should be shared or separated. By introducing hierarchical partitions, we can capture heterogeneity at multiple levels: sample-level variation within each context, and task-level switching across contexts. This perspective connects classical partition-based models with multi-task learning, highlighting how explicit splits of context define where parameters should be shared versus differentiated (Figure {@fig:context-splits}). diff --git a/content/manual-references.json b/content/manual-references.json index d523975..cd0103d 100644 --- a/content/manual-references.json +++ b/content/manual-references.json @@ -176,5 +176,20 @@ "volume": "22", "issued": {"date-parts": [[2009]]}, "URL": "https://proceedings.neurips.cc/paper_files/paper/2009/hash/7bcdf75ad237b8e02e301f4091fb6bc8-Abstract.html" + }, + { + "id": "hothorn2015partykit", + "type": "article-journal", + "title": "partykit: A Modular Toolkit for Recursive Partytioning in R", + "author": [ + {"family": "Hothorn", "given": "Torsten"}, + {"family": "Zeileis", "given": "Achim"} + ], + "container-title": "Journal of Machine Learning Research", + "volume": "16", + "issue": "118", + "page": "3905-3909", + "issued": {"date-parts": [[2015]]}, + "URL": "https://jmlr.org/papers/v16/hothorn15a.html" } ]