Bayes-THIS: Bayesian Hypergraph Inference

• Bayes-THIS is a Bayesian extension of the Taylor-based THIS method that infers higher-order interaction structures from dynamical time-series data.
• It replaces fixed-threshold sparse regression with automatic relevance determination, enabling uncertainty-aware coefficient selection and robust inference.
• Empirical results show that Bayes-THIS outperforms the original THIS approach in challenging regimes such as scarce, noisy, or ill-conditioned data.

Searching arXiv for Bayes-THIS and the original THIS method. arXiv_search(query="Bayes-THIS Taylor-based Hypergraph Inference using SINDy THIS hypergraph", max_results=10, sort_by="relevance") arXiv_search query: "Bayes-THIS" search_arxiv({"query":"Bayes-THIS hypergraph inference SINDy", "max_results": 5}) Bayes-THIS is a Bayesian extension of Taylor-based Hypergraph Inference using SINDy (THIS) for reconstructing higher-order interaction structure from dynamical time-series data. It is designed for settings in which the observed trajectories are scarce, noisy, or concentrated in a small region of state space, so that sparse regression on a Taylor library becomes weakly constrained or ill-conditioned. The method preserves the original THIS representation of hyperedges through Taylor coefficients, but replaces fixed-threshold sparse regression with sparse Bayesian regression using automatic relevance determination (ARD), explicitly models residual variance, and introduces an uncertainty-aware workflow based on posterior predictive checks and credible-interval pruning. At the same time, it inherits a structural limitation of the Taylor representation: certain spurious lower-order edges generated by higher-order interactions are fundamentally non-identifiable within the framework (Tang et al., 5 May 2026).

1. Conceptual position within hypergraph inference

Bayes-THIS addresses the inverse problem of inferring pairwise and higher-order interaction structure from observations of node dynamics evolving on a hypergraph. The target systems are modeled as having interaction tensors $A^{(p)}=\{a_{i_1,\dots,i_p}^{(p)}\}$, with the paper focusing primarily on up to third-order interactions. The underlying motivation is that many real dynamical systems are not adequately described by pairwise couplings alone, yet the available trajectories are often too limited or too noisy for reliable direct recovery of higher-order structure (Tang et al., 5 May 2026).

The original THIS method casts the problem into the SINDy paradigm: construct a monomial library from observed states, regress the observed derivatives onto that library, and interpret nonzero Taylor coefficients as evidence for specific hyperedges. Bayes-THIS does not alter this representational layer. Its contribution is inferential. In place of sequential thresholded least squares (STLS) or LASSO with fixed sparsity tuning, it uses sparse Bayesian regression with ARD. This changes the procedure from threshold-based coefficient selection to posterior inference with adaptive, term-wise shrinkage and explicit uncertainty quantification (Tang et al., 5 May 2026).

This design matters most in the regimes where THIS is fragile. The paper identifies three such regimes: scarce data, high observation noise, and trajectories concentrated near lower-dimensional regions of state space. In each case, the regression matrix becomes underdetermined, noisy, or nearly collinear, and a global hard threshold becomes difficult to tune. Bayes-THIS is proposed precisely as a robustness-oriented Bayesian upgrade for those settings (Tang et al., 5 May 2026).

2. Dynamical model and Taylor encoding of hyperedges

The starting point is a hypergraph-coupled dynamical system

$$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$

Here the coefficients $a_{ij}^{(2)}$ and $a_{ijk}^{(3)}$ encode pairwise and triadic interactions. The key observation inherited from THIS is that expanding $F_i$ about a reference point $\mathbf{x}^*$ yields a Taylor series whose nonzero monomial coefficients correspond to interaction structure:

$$\dot{x}_i = F_i(\mathbf{x}^*) + \sum_{j=1}^n \partial_j F_i(\mathbf{x}^*) \Delta x_j + \frac{1}{2}\sum_{j,k=1}^n \partial_{j,k} F_i(\mathbf{x}^*) \Delta x_j \Delta x_k + \cdots,$$

with $\Delta x_j = x_j - x_j^*$ (Tang et al., 5 May 2026).

Under the third-order assumption emphasized in the paper, a nonzero mixed derivative $\partial_{j,k}F_i(\mathbf{x}^*)$ indicates a triadic interaction among $i,j,k$. Operationally, this means that hypergraph reconstruction can be recast as sparse identification of active monomials in a polynomial library. Given data matrix $$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$0, the library is written as

$$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$1

and the noiseless linear map is

$$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$2

Nonzero entries of $$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$3 identify active monomials and therefore candidate edges or hyperedges (Tang et al., 5 May 2026).

The representational strength of THIS and Bayes-THIS is thus the same: both translate nonlinear higher-order dynamics into a sparse regression problem over a Taylor basis. Bayes-THIS changes how that sparse inverse problem is regularized and interpreted, not what it means geometrically.

3. Bayesian sparse regression with automatic relevance determination

For a given node, Bayes-THIS assumes the noisy linear model

$$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$4

The corresponding likelihood is

$$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$5

The residual variance is therefore part of the inferential model rather than an external nuisance ignored by thresholding (Tang et al., 5 May 2026).

Sparsity is imposed through an ARD prior. Each coefficient receives an independent zero-mean Gaussian prior with its own precision:

$$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$6

Large $$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$7 implies strong shrinkage of $$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$8 toward zero. Each $$\dot{x}_i = F_i(\mathbf{x}) = f_i(x_i) + \sum_{j=1}^n a_{ij}^{(2)} g^{(2)}(x_i, x_j) + \sum_{j,k=1}^n a_{ijk}^{(3)} g^{(3)}(x_i,x_j,x_k) + \cdots.$$9 in turn has a Gamma hyperprior,

$a_{ij}^{(2)}$0

with the minimally informative choice $a_{ij}^{(2)}$1 used in the paper to make the prior scale-invariant. If the noise variance is unknown, the paper reparameterizes it as $a_{ij}^{(2)}$2 and re-estimates it iteratively by maximizing evidence (Tang et al., 5 May 2026).

Because direct integration over $a_{ij}^{(2)}$3 is intractable, the method uses type-II maximum likelihood. The marginal likelihood, or log-evidence, is

$a_{ij}^{(2)}$4

with

$a_{ij}^{(2)}$5

The ARD mechanism removes terms by driving $a_{ij}^{(2)}$6, so that the corresponding prior on $a_{ij}^{(2)}$7 collapses toward a spike at zero. The paper describes this as an approximation to a spike-and-slab prior with better tractability than MCMC (Tang et al., 5 May 2026).

Conditioned on $a_{ij}^{(2)}$8 and $a_{ij}^{(2)}$9, the posterior over coefficients is Gaussian:

$a_{ijk}^{(3)}$0

Its mean and covariance are

$a_{ijk}^{(3)}$1

Bayes-THIS uses $a_{ijk}^{(3)}$2 as the point estimate and retains $a_{ijk}^{(3)}$3 for uncertainty quantification (Tang et al., 5 May 2026).

4. Uncertainty-aware workflow and coefficient selection

A central feature of Bayes-THIS is that it is presented as a workflow rather than only an estimator. The paper describes three stages: build the Taylor monomial library, assess whether the data support higher-order inference beyond a pairwise model, then fit the full sparse Bayesian model and prune coefficients by posterior uncertainty (Tang et al., 5 May 2026).

The posterior predictive check is intended to answer a restricted question: whether the available data contain enough higher-order signal to justify inference beyond pairwise structure. The library is partitioned as

$a_{ijk}^{(3)}$4

where $a_{ijk}^{(3)}$5 contains pairwise terms and $a_{ijk}^{(3)}$6 contains triadic terms. After fitting the pairwise-only model, the method draws posterior samples $a_{ijk}^{(3)}$7 and computes observed residuals

$a_{ijk}^{(3)}$8

The triadic design matrix is orthogonally projected away from the pairwise subspace via

$a_{ijk}^{(3)}$9

and the projector

$F_i$0

defines the discrepancy statistic

$F_i$1

Replicated residuals $F_i$2 yield

$F_i$3

and the posterior predictive $F_i$4-value is

$F_i$5

A large $F_i$6-value means the residuals look like noise in triadic directions and the data do not reliably support higher-order inference; a small $F_i$7-value means residual structure beyond the pairwise model remains detectable. The paper is explicit that this is a filter for unsafe inference, not a guarantee of successful recovery (Tang et al., 5 May 2026).

Coefficient selection is also uncertainty-aware. Bayes-THIS does not primarily retain terms by thresholding $F_i$8. Instead, it examines the conditional posterior of each coefficient given the others fixed at posterior means. If the posterior covariance is partitioned as

$F_i$9

then the conditional variance is

$\mathbf{x}^*$0

The equal-tailed $\mathbf{x}^*$1-credible interval is

$\mathbf{x}^*$2

A coefficient is retained if and only if zero lies outside this interval. The intended effect is adaptivity: when uncertainty is high because of noise, small sample size, or collinearity, the interval widens automatically. The paper also notes that this pruning step is not fully Bayesian, because it conditions on point estimates of other coefficients and hyperparameters rather than integrating them out (Tang et al., 5 May 2026).

5. Empirical operating regimes and observed advantages

The empirical results reported for Bayes-THIS focus on the regimes in which sparse regression is most difficult. In a Kuramoto-like model with pairwise and triadic interactions, Bayes-THIS outperforms THIS over most of the $\mathbf{x}^*$3-versus-$\mathbf{x}^*$4 parameter space. The paper states that Bayes-THIS maintains high AUROC over nearly the whole range, whereas THIS is highly sensitive to the choice of STLS threshold $\mathbf{x}^*$5, and no single $\mathbf{x}^*$6 performs well in both low-noise and high-noise regimes. The largest gains occur when data are scarce and noise is substantial (Tang et al., 5 May 2026).

The same pattern appears in triadic precision and recall. Bayes-THIS is reported to be especially better in the sparse regime with few true triads and many false candidates. At very high noise, neither method recovers triads well, and the performance gap narrows. This is a bounded claim: Bayes-THIS improves robustness, but the paper does not represent it as overcoming extremely adverse signal-to-noise regimes (Tang et al., 5 May 2026).

Ill-conditioning is another major test case. When trajectories concentrate near a low-dimensional manifold, columns of $\mathbf{x}^*$7 become nearly dependent. In the Kuramoto experiments this occurs as coupling increases and synchronization tightens. Under such conditions, STLS coefficients fluctuate enough that true terms may fall below threshold while false terms cross it. Bayes-THIS continues to outperform THIS because its shrinkage is term-wise and its pruning rule accounts for posterior variance. Even so, the paper notes that under severe ill-conditioning Bayes-THIS can still develop broadened or even bimodal coefficient posteriors. The improvement is therefore robustness, not immunity (Tang et al., 5 May 2026).

The method also scales more favorably as the number of nodes grows and the candidate library expands combinatorially. As $\mathbf{x}^*$8 increases from 7 to 30, Bayes-THIS continues to outperform THIS in noisy, data-limited settings. A countervailing effect appears in dense true hypergraphs: because ARD imposes a sparsity bias, Bayes-THIS can be slightly worse than THIS when many weak coefficients are genuinely present. The paper characterizes this effect as small and notes that the gap largely disappears if THIS is made less aggressively sparse. This suggests that Bayes-THIS is best matched to sparse or moderately sparse discovery problems, which the paper treats as the primary use case (Tang et al., 5 May 2026).

6. Structural non-identifiability and methodological limits

The principal limitation of Bayes-THIS is not statistical but representational. The paper shows that higher-order interactions generically contribute to lower-order Taylor coefficients. As a result, a genuine triadic interaction can induce nonzero pairwise monomial coefficients in the Taylor expansion. When this happens, spurious pairwise edges may be indistinguishable from genuine lower-order interactions using only the Taylor coefficients recovered by THIS or Bayes-THIS (Tang et al., 5 May 2026).

The decisive quantity is cross-order degree correlation (DC), defined informally in the paper as the correlation between pairwise degree $\mathbf{x}^*$9 and triadic degree $$\dot{x}_i = F_i(\mathbf{x}^*) + \sum_{j=1}^n \partial_j F_i(\mathbf{x}^*) \Delta x_j + \frac{1}{2}\sum_{j,k=1}^n \partial_{j,k} F_i(\mathbf{x}^*) \Delta x_j \Delta x_k + \cdots,$$0. When cross-order DC is high, triadic edges tend to sit on nodes that already carry pairwise edges, as in simplicial complexes. In that case, the lower-order contributions induced by triads overlap with genuine pairwise signal and are less disruptive. When cross-order DC is low, higher-order interactions concentrate on nodes or pairs that lack lower-order connections. Then the downward contribution from triads produces spurious pairwise coefficients on absent pairwise edges (Tang et al., 5 May 2026).

The paper treats this as a structural non-identifiability of the Taylor-based framework itself. It cannot be resolved by choosing Bayes-THIS over STLS, by retuning thresholds, or by post-processing the recovered coefficients. Even credible-interval pruning fails here: if the posterior strongly supports a spurious pairwise coefficient, the Taylor representation contains no information indicating that it should instead be attributed to a triadic interaction. Empirically, as cross-order DC decreases, pairwise precision drops sharply, false positives increase, recall can remain high, and pairwise AUPRC worsens because true and triad-induced false pairwise coefficients overlap across thresholds. The paper emphasizes that this persists even with good data and low noise, showing that the failure mode is not merely a consequence of finite-sample uncertainty (Tang et al., 5 May 2026).

Additional caveats follow from the modeling assumptions. Bayes-THIS still relies on a truncated Taylor expansion, so truncation error lies outside the Bayesian layer. It assumes independent Gaussian derivative noise; correlated noise may require whitening or more sophisticated modeling. Its posterior predictive check detects only one failure mode, namely sampling too close to the expansion point, and cannot diagnose sampling too far away, where the Taylor approximation itself degrades. These constraints delimit the method’s scope. A plausible implication is that Bayes-THIS should be understood not as a universal hypergraph reconstruction procedure, but as a Bayesian regularization and uncertainty-quantification framework for a specific Taylor/SINDy representation whose strongest and weakest points are both inherited from that representation (Tang et al., 5 May 2026).