In this work, we formulate a systematic expectation-value framework for dynamical systems whose probability densities evolve according to linear partial differential equations, such as the Fokker-Planck and Liouville equations. The approach is based on expectation-calculus identities associated with the Fluctuation-Dissipation Theorem and the Conjugate Variables Theorem, allowing the derivation of evolution equations directly for arbitrary observables and fluctuations without explicitly solving the full probability-density equation. The resulting relations provide a classical Ehrenfest-type formulation for observable dynamics and fluctuations under linear probability-density evolution. While the resulting equations are not closed in general, since they typically involve higher-order moments, correlations, or derivatives, the formalism offers a unified operational framework for studying observable dynamics under suitable approximations or closure assumptions. We illustrate the procedure with examples involving Fokker-Planck and Liouville dynamics and discuss the scope, limitations, and possible applications of the framework in nonequilibrium statistical mechanics. In particular, we emphasize that the method is intended as a systematic observable-based formulation for systems governed by linear evolution equations, rather than as a universal closure scheme for arbitrary nonequilibrium dynamics.
The morphing glide aircraft (MGA) can adapt to complex environments and mission requirements by altering its aerodynamic configuration through the jettisoned wings, which face the challenges in dynamic and aerodynamic characteristics variation, as well as lumped uncertainty. To deal with the above issues, a neural network (NN) parameter identification-based prescribed-time adaptive control for MGA is proposed. Firstly, a time-scale function is proposed, which avoids the singularity and the unrealistic issue of unbounded growth in control effort. Further, a fractional-power prescribed-time Lyapunov stability theorem is established, which overcomes the limitation of conventional theorems in analyzing robust sliding mode control with non-smooth control terms, providing a theoretical foundation for the design and stability analysis of prescribed-time fractional-power sliding mode controllers. Then, for the issue of aerodynamic parameter uncertainty, an NN parameter identification method is proposed. Based on this, a prescribed-time adaptive sliding mode control of MGA is proposed to guarantee the controller error convergence in the prescribed-time, independent of initial conditions and control parameters. Finally, the proposed algorithm is employed to achieve attitude tracking for the MGA, and the effectiveness is illustrated.
Modern quantum physics now enables control of quantum systems at the level of individual trajectories, opening a new frontier that links quantum information theory, quantum many-body physics, and quantum thermodynamics, and uncovers novel non-equilibrium phenomena such as deep thermalization and measurement-induced entanglement. However, a central challenge remains: their characterization relies on measuring nonlinear properties of individual quantum states, a task tantamount to fine-grained cloning of a quantum ensemble. Here, the fundamental laws governing the cloning of quantum ensembles are investigated. First, a general no-cloning theorem for arbitrary ensembles is established from an information-theoretic perspective, even assuming multiple copies of the ensemble's purification. It is then shown that this barrier can be unexpectedly circumvented for physical ensembles generated by finite-time evolutions. Nevertheless, these tasks are proven to remain computationally intractable, even when the full circuit description of state preparation is known. This stands in sharp contrast to the conventional no-cloning theorem, which relies on the state being unknown. Together, these results establish new fundamental principles of quantum mechanics, reveal intrinsic trade-offs among sample complexity, computational complexity, and quantum measurements, and highlight the necessity of problem-specific strategies for probing measurement-induced quantum phenomena.
Operator learning for parametric partial differential equations (PDEs) has emerged as a transformative paradigm, with physics-informed neural networks (PINNs), Deep Operator Networks (DeepONet), and Fourier Neural Operators (FNO) demonstrating remarkable capabilities. However, these methods face a critical tradeoff: data-driven operators (DeepONet, FNO) require 103-104 expensive solution snapshots for training, while physics-informed approaches (PINNs) lack rigorous convergence guarantees due to non-convex soft constraints computed via automatic differentiation. This creates a fundamental gap: no existing method provides provable O(hk+1-m) approximation-capacity bounds while simultaneously requiring zero training data and maintaining fast multi-query inference. To address these limitations, we propose Convolutional Neural Operators with Physics-Encoded Kernels (CNO-PEK), a novel operator learning framework that pre-computes differential operators as spatially-adaptive convolutional kernels through the Physics-Informed Kernel Field (PIKF). Unlike data-driven operators requiring thousands of solution snapshots [Formula: see text] or PINNs computing residuals via automatic differentiation at random collocation points, our method encodes the governing physics directly into the network architecture by analytically deriving kernels from polynomial consistency conditions. This architectural design provides provable O(hk+1-m) approximation-capacity bounds matching classical finite element theory (Theorem 1, subject to polynomial representability; practical convergence of the trained network additionally requires optimization to a global minimum, as stated in Proposition 1)), while simultaneously leveraging neural networks' superior approximation capabilities without requiring any solution training data. Through comprehensive validation on 1D-3D benchmark problems and systematic comparison with PINNs, DeepONet, and classical methods, we demonstrate: (1) O(hk+1-m) approximation-capacity bounds (Theorem 1) confirmed empirically, (2) 60.9% systematic error reduction compared to linear finite elements (mean error 1.72% vs. 4.40%) without requiring training data, (3) physics-driven zero-shot parameter generalization (trained once at μ=2.5 with no solution data at any μ ∈ [0, 5]) across 1000 unseen parameter values, with 100% of cases achieving  < 3% error, (4) computational speedup of 6.9-13.6 ×  for multi-query scenarios beyond a break-even point of approximately 843 queries. We position CNO-PEK as a theoretically grounded operator learning framework that bridges classical numerical analysis and modern deep learning, particularly suited for parametric studies, uncertainty quantification, and real-time applications where approximation-capacity guarantees and data efficiency are critical.
This paper proposes DHCRWOA, a stage-specific extension of the Whale Optimization Algorithm (WOA) designed to improve its exploration-exploitation transition and reduce stagnation around suboptimal leaders. Five modules are introduced, each targeting a single WOA search phase: Good Nodes Set (GNS) initialization, adaptive parameter control, statistical-guided exploration, Cauchy-based local exploitation, and Rayleigh-weighted spiral updating. On the CEC2005 benchmark suite at 30 dimensions, DHCRWOA achieves the best average rank (2.13) among ten algorithms and the best or tied-best mean on 22 of 23 functions, with no baseline outperforming it on more than one function. At 50 and 100 dimensions, DHCRWOA maintains the top Friedman rank (1.69 and 1.62, respectively) at average runtimes of 0.52 s and 1.03 s per run. A five-variant ablation study confirms that weakening the statistical-guided exploration or Cauchy-based exploitation produces the largest rank degradation (from 2.35 to 4.09 and 4.70, respectively), while all five modules contribute to overall performance. Five constrained engineering design problems further confirm competitive results on practical optimization tasks. Theoretical analysis establishes the relationship between DHCRWOA and standard WOA, including a perturbation bound on the adaptive parameter ([Formula: see text]), boundedness of the population trajectory, monotonicity of the best-so-far process, and a global reachability theorem.
The phase of spins in the quasi-two-dimensional (q2D) XY model has emerged as a topic of significant interest across multiple subfields of physics. Conventional wisdom, rooted in the Mermin-Wagner theorem and supported by existing paradigms, asserts that true long-range (LR) order is prohibited in q2D systems with continuous symmetries and short-range (SR) interactions. In this Letter, we propose a strictly SR q2D XY model defined on a plane perpendicularly intersected by a group of parallel planes, where each plane consists of XY spins coupled via nearest-neighbor interactions. Through large-scale Monte Carlo simulations complemented by finite-size scaling analysis, we establish the complete phase diagram of the setup. A LR ordered phase emerges in the q2D model when the spins on the parallel planes develop a Berezinskii-Kosterlitz-Thouless critical phase. The LR ordered phase is anisotropic: true LR correlations develop exclusively along the direction of the intersection lines, while the perpendicular direction exhibits quasi-long-range order. Furthermore, the LR order exhibits Goldstone-mode physics. Our findings reveal a mechanism for stabilizing LR order in low-dimensional systems with continuous symmetries, thereby establishing a new platform for studying exotic superfluidity.
Dengue fever, a rapidly expanding mosquito-borne viral disease, poses a significant global public health challenge due to complex transmission dynamics influenced by human behavior and environmental factors and continues its relentless global expansion, with approximately 390 million annual infections straining health systems across endemic regions, yet conventional models persistently overlook the critical feedback loop wherein rising case numbers trigger public awareness that subsequently alters transmission behavior. This study develops and rigorously analyzes a novel mathematical framework that elevates prevalence-driven public awareness from an absent or static parameter to a dynamic state variable, fundamentally advancing beyond existing approaches that unrealistically assume innate awareness at birth. We construct a deterministic compartmental model using seven ordinary differential equations, stratifying human populations into unaware susceptible, aware susceptible, infected, and recovered classes coupled with susceptible and infected mosquito compartments, while implementing a biologically grounded acquisition mechanism where all newborns enter the unaware class and gain awareness through baseline educational programs at rate ν and prevalence-driven campaigns at rate [Formula: see text]. The methodological approach proceeds through four integrated phases: derivation of the basic reproduction number [Formula: see text] via the next-generation matrix method with rigorous stability analysis of the disease-free and endemic equilibria; comprehensive local sensitivity analysis using elasticity indices and global sensitivity analysis employing Partial Rank Correlation Coefficients with [Formula: see text] Latin Hypercube samples to identify the most influential transmission parameters; formulation of an optimal control problem with four time-dependent interventions comprising vector control [Formula: see text], surge awareness campaigns [Formula: see text], treatment acceleration [Formula: see text], and personal preventive measures [Formula: see text]; and numerical solution of the optimality system using Pontryagin's maximum principle with forward-backward sweep implementation. The principal contributions include the first integrated framework explicitly linking prevalence-driven awareness dynamics to dengue transmission with realistic newborn acquisition pathways, rigorous analytical results encompassing [Formula: see text] characterization and stability theorems, and a complete optimal control solution demonstrating synergistic effects among four simultaneous interventions. Key findings reveal that mosquito mortality rate [Formula: see text] represents the most influential parameter (elasticity [Formula: see text], PRCC [Formula: see text], [Formula: see text]), awareness parameters exhibit statistically significant negative effects (α PRCC [Formula: see text], ν PRCC [Formula: see text], [Formula: see text]), and optimal control analysis demonstrates that single interventions achieve 45-[Formula: see text] peak infection reduction while the full four-control combination attains [Formula: see text] reduction, transforming an outbreak peak of approximately [Formula: see text] cases to below 150 cases. These results provide a quantitative, mathematically rigorous roadmap for resource allocation in dengue-endemic regions, establishing that integrated multi-pronged intervention portfolios yield superior outcomes that transcend the sum of their individual components.
Three-component spinor Bose-Einstein condensates (BECs) provide an ideal platform to systematically investigate nonlinear wave dynamics under non-zero background. Using the Darboux transformation, we derive exact breather and rational solutions in which only one component maintains a zero background, while the other two reside on finite backgrounds. Based on the residue theorem associated with fourth-order singularities, we further derive physical spectra of these exact solutions. Our analysis shows that the energy-transfer feature identified in the spectra should be understood as the spectral manifestation of spin-exchange-induced population redistribution from the two non-zero-background components to the zero-background component. This redistribution causes the zero-background component to exhibit breather-like density characteristics. On the other hand, for rational solutions, the components with non-zero background exhibit classical rogue wave behavior, while the component with zero-background forms a novel valley-free double-peak structure. The numerical simulations show that these rational solutions are quite stable in their initial stages. In addition, we also derived high-order rational solutions and their spatial distribution patterns. Our results provide deeper insights into the dynamics of localized waves and enrich nonlinear excitations in coupled multi-component BEC systems.
Bell's theorem rules out developing a locally causal theory to describe quantum phenomena. Many take this to imply that any model of quantum entanglement must employ variables (called beables by Bell) which follow nonlocal rules, even though signaling is local. The alternative is to adopt an all-at-once (block universe) approach, with beables which may depend on both past and future inputs, even though signaling is causal. Within this lenient-causality approach (a.k.a. retrocausal), simple cases of entanglement have been successfully described by locally mediated stochastic toy models, i.e., toy models which are local in a sense which generalizes Bell's local causality. Developing a widely applicable reformulation of quantum mechanics along these lines is a grand challenge. This work presents a general framework for such models and theories, and identifies the corresponding ontic and epistemic states. The epistemic state is closely analogous to the quantum state, yielding an explanation for the collapse of the wavefunction. In the case of the models of the framework, it is clear what the information is about. The expression for the empirically verifiable predictions of the models in terms of the ontic and epistemic states displays remarkable parallels to the Born rule. A toy-model example is discussed.
The Stieltjes moment problem is studied in a new framework within the general Gelfand-Shilov spaces defined via weight sequences. The novelty consists of allowing for a naturally larger target space for the moment mapping, which sends a function to its sequence of Stieltjes moments. The motivation comes from a recent version of the Borel-Ritt theorem, concerning the surjectivity of the Borel mapping in Carleman-Roumieu ultraholomorphic classes in sectors, whose defining weight sequence is subject to the condition, weaker than derivation closedness, of having shifted moments. The injectivity and surjectivity of the moment mapping in this new setting is studied and, in some cases, characterized. Finally, results are provided for general weight sequences of fast and regular enough growth when the condition of shifted moments fails to hold.
Algorithmic complexity is a foundational notion in theoretical computer science, but its incomputability has led to two families of practical estimators: compression-based and program-execution-based (e.g., the Coding Theorem Method, CTM). Despite widespread use, the correspondence between these paradigms remains poorly understood. We present a systematic comparative framework that uses the Block Decomposition Method (BDM) to extend CTM-based estimates to longer strings, enabling direct comparison with compression-based estimators across multiple computational models. A control estimator (BDMId) isolates the contribution of block structure from algorithmic information, providing a rigorous baseline for interpreting correlations. Our results show that cross-paradigm correlations are weak and decrease systematically as model resolution decreases; for the lowest-resolution model, correlations are essentially null. In long strings, per-length correlations vanish, while global correlations appear high but are largely explained by the control estimator, indicating that they are driven primarily by trivial length effects rather than shared sensitivity to algorithmic structure. Crucially, for low-resolution models, BDMId outperforms BDM itself, indicating that the inclusion of CTM information does not improve-and may even reduce-agreement with compression-based estimators. These findings suggest that compression-based and program-execution-based estimators capture fundamentally different aspects of structure. Rather than invalidating either approach, this work provides a systematic methodology for assessing cross-paradigm correspondence and highlights the importance of explicit controls in empirical comparisons of algorithmic complexity.
Fission and fusion of lipid membranes are ubiquitous shape transformations in living cells, necessary for maintaining the shape of organelles and enabling a host of transport processes between them. These topology-changing events involve curvature-elastic free energy changes proportional to the Gaussian curvature modulus, κ[over ¯]. However, precisely because of its topological nature, the value of this modulus is exceptionally hard to determine. Here we propose a novel method to computationally determine κ[over ¯] that involves simulating triply periodic minimal surfaces. We measure their excess curvature energy E and infer the associated free energy F via a thermodynamic argument that rests on the strong temperature dependence of E and the relatively weak temperature dependence of E/F. Our approach is unique in that it circumvents the complications arising in alternative procedures that require an open membrane edge to break the constraints of the Gauss-Bonnet theorem. As an illustration, we determine κ[over ¯] for a coarse-grained lipid model over a range of temperatures and lipid shapes. We show that the observed changes of both the Gaussian curvature modulus as well as its ratio to the ordinary bending modulus follow widely held expectations.
The aurochs, the wild ancestor of domestic cattle, was a keystone herbivore in Late Pleistocene Eurasian ecosystems and a major prey species for Palaeolithic hunter-gatherers. Despite its significance, the genetic structure of aurochs populations that survived the Last Glacial Maximum (LGM) remains poorly understood, especially in southeastern Europe. Here, we present the first directly dated ancient genomes of aurochs from the eastern Adriatic region, recovered from the Upper Palaeolithic site of Šandalja (Istria, Croatia). Two female individuals, dated to approximately 14,800-14,200 and 11,800-11,400 calibrated years before present, were sequenced for low-coverage whole genomes and near-complete mitochondrial genomes. Bayesian phylogenetic analyses and median-joining network reconstruction place both specimens within mitochondrial haplogroup P, the dominant European aurochs lineage. However, they do not cluster within the main P sub-haplogroup observed in most ancient aurochs samples and in modern cattle carrying P-type mitochondrial lineages. Instead, one specimen is placed within an 'alternative' P sub-haplogroup, whereas the position of the other appears more isolated and should be interpreted cautiously, as it may be influenced by limited sequence coverage and the resulting uncertainty in phylogenetic placement. At the nuclear genomic level, the two Šandalja aurochs show affinity to Late Pleistocene and Early Holocene aurochs from Italy. Although based on a limited number of specimens, this pattern is consistent with possible genetic connectivity across the Adriatic region, potentially associated with the now-submerged Great Adriatic Plain (GAP). Overall, our results suggest regional structure among Late Pleistocene aurochs, potentially associated with the exposed Adriatic Plain as a refugium and dispersal corridor between the Apennine and Balkan Peninsulas. By filling a major geographic and temporal gap in the aurochs genomic record, this study highlights the Adriatic Basin as a potentially overlooked centre of Pleistocene megafaunal diversity and refines models of postglacial recolonization and cattle evolutionary history.
The infant microbiome undergoes rapid changes in composition over time and is associated with long-term risks of conditions such as immune strength, allergy, asthma, and other health outcomes. Modeling the associations between exposures or treatments and microbial composition over time is essential for understanding the factors that drive these changes. Estimating these temporal dynamics has several challenges including repeated measures, overdispersion, compositionality, high-dimensional parameter spaces, and zero-inflation. Many longitudinal regression models used in human microbiome research assume constant effects over time that cannot capture time-varying or functional effects of exposures, ignore the compositional structure of the data by modeling each taxon separately, and are not equipped to handle potential zero-inflation. Dirichlet-multinomial (DM) regression models inherently accommodate overdispersion and the compositional structure of the data and have been extended to account for excess zeros. However, existing DM-based regression models are unable to additionally handle repeated measures designs. To fill this gap, we propose a functional concurrent zero-inflated Dirichlet-multinomial regression model which is designed to model time-varying relations between observed covariates and microbial taxa while accounting for zero-inflation, compositionality, and repeated measures. Through simulation, we demonstrate that the model can accurately estimate the underlying functional relations and scale to large compositional spaces. We apply our model to investigate time-varying associations between infant microbiome composition and observed covariates during the 11-wk postnatal period. We found that $ \boldsymbol{\alpha} $-diversity (ie the diversity of the microbiome within an individual) is positively associated with a higher gestational age and percentage of breast milk in the diet. We provide an accompanying R package and shiny app to implement the method and generate plots.
The Cape sea urchin, Parechinus angulosus, is a widely distributed keystone species that inhabits intertidal and subtidal ecosystems along the South African coastline. Despite its importance as an ecosystem engineer, its phylogenetic placement and mitochondrial genomic (hereafter mitogenome) variation remain poorly understood. In the current study, we present the first complete mitogenome for this species, assembled from long-read sequences generated on the Oxford Nanopore sequencing platform, and investigate its phylogenetic placement among other sea urchin species using a combination of Bayesian Inference and Maximum-Likelihood methods. A circular genome of 15 722 bp, with an average coverage of 159, comprising 13 protein-coding genes, two rRNAs and 22 tRNAs, was assembled de novo. Phylogenetic reconstructions based on 13 protein-coding genes recovered Paracentrotus lividus as the sister taxon of P. angulosus, and these two species formed a monophyletic clade with Loxechinus albus and Sterechinus neumayeri within Camaradont sea urchins. Using the mitogenome assembly as a template, an additional set of 29 cox1 sequences was mined from publicly available genomic sequences. These revealed that Cape sea urchins maintain substantial mitogenomic variation across their distribution range, expressed predominantly as low-frequency haplotypes. This study demonstrates that the Cape sea urchin is genetically distinct within the order Camarodonta and exhibits considerable variation in the cox1 gene across coastal habitats of southern Africa. Furthermore, the identification of a large number of low-frequency haplotypes may indicate population expansion or ongoing purifying selection.
Imported malaria is a critical obstacle to achieving elimination in low transmission settings, but importation classification tools combining human mobility and parasite genomics are lacking. A Bayesian model combining epidemiological, human mobility, and parasite genetic data was developed to estimate malaria importation and geographic origins of Plasmodium falciparum cases. Using microhaplotype-based genetic relatedness from 1605 samples across nine Mozambican provinces in 2022, the study focused on two low-transmission districts in the south: Magude and Matutuine. Parasites from southern Mozambique showed lower genetic relatedness to those from northern/central regions (0.021) than the national average (0.034, p<0.001), indicating limited connectivity. Overall, 42% (88/207) of infections in these districts were classified as imported, mainly originating from Inhambane province (63% [55/88]). Imported cases showed higher parasite complexity than local ones (odds ratios [OR] = 1.3). Importation rates differed markedly between districts - Matutuine (48.60%, 87/179) was far more affected than Magude (10.71%, 3/28) - highlighting the need for localised rather than uniform elimination strategies. In Matutuine, importation appears to be actively sustaining transmission, suggesting that reducing malaria burden in source regions (particularly Inhambane) and targeting travellers from central and northern Mozambique would have the greatest elimination impact.
Tuberculosis (TB) remains a major public health challenge in Thailand, a high-burden country undergoing both epidemiological transition and pandemic-related disruption. This study examined temporal and spatial patterns of age-standardized TB incidence from 2012 to 2023 across Thailand's 13 health regions and 77 provinces. This nationwide ecological study used annual TB case notifications (2012-2023) for all 77 Thai provinces from Thailand's National Disease Surveillance System (Report 506), Bureau of Epidemiology, with mid-year and age-stratified provincial populations. Age-standardized incidence rates (ASR) were calculated using the WHO World Standard Population (2000-2025). Temporal trends were assessed by Joinpoint regression with Bayesian-Information-Criterion model selection (maximum 2 joinpoints) and validated by generalized additive models. Spatial clustering was evaluated by Global Moran's I and Local Indicators of Spatial Association (LISA) under queen contiguity, with sensitivity analyses across alternative weights and Benjamini-Hochberg false-discovery-rate correction. Provinces were classified into long-term trend categories combining effect size (AAPC) and significance, and COVID-19 impact was quantified by 2019-2023 ASR percent change. National TB incidence declined substantially from 2012 to 2023, although marked regional heterogeneity persisted. Five health regions showed the strongest long-term reductions: region 3 [AAPC: - 20.51, 95% confidence interval (CI): - 33.00 to - 13.77], region 9 (- 19.42, 95% CI: - 28.98 to - 8.57), region 4 (- 17.53, 95% CI: - 23.64 to - 12.89), region 11 (- 11.79, 95% CI: - 22.16 to - 0.04), and region 13 (- 11.19, 95% CI: - 22.44 to - 3.74). Region 6 showed an early decline followed by stabilisation, and region 12 showed a mid-period increase before a post-2019 reduction. Thailand has achieved substantial overall reductions in TB incidence over the past decade, but pronounced regional and provincial disparities persist. Localized hot-spots, biphasic regional trajectories that may reflect surveillance and programmatic transitions, and compound vulnerability during the COVID-19 period underscore the need for geographically targeted monitoring, equitable resource allocation, and pro-poor interventions to sustain progress toward TB elimination.
Parkinson's disease (PD) is a complex neurodegenerative disorder with a significant genetic component. While genome-wide association studies (GWAS) have been instrumental in identifying genetic variants associated with PD, the reliance on large sample sizes and population-level analyses may overlook variants with lower minor allele frequencies or individual-specific relevance. Individualized Bayesian Inference (IBI) offers a promising method to complement GWAS by identifying and prioritizing candidate genetic markers at both the individual and patients-like-me subgroup levels. This study evaluates the application of IBI to PD genetics, using GWAS as a baseline for comparison. We analyzed genetic data from the Fox Insight online study, including 8840 individuals (8585 PD cases and 255 controls). IBI prioritized variants that were not detected or were ranked substantially lower by GWAS, including variants within or near genes with prior PD association. The top 200 IBI SNPs showed stronger predictive performance in ANN models (AUC = 0.79) than the top 200 GWAS SNPs (AUC = 0.72), providing complementary support for the utility of IBI-based prioritization in this cohort. Notably, IBI highlighted variants with lower minor allele frequencies that GWAS did not detect. This study demonstrates the utility of IBI as a complementary tool for prioritizing PD-related candidate variants and genes for further investigation.
This article introduces a pyramid-based multiresolution Bayesian framework for high-resolution behavioral analysis from sparse data. By restricting covariance modeling to the coarsest layer of a parameter pyramid while using a difference pyramid for fine-scale refinement, the framework overcomes major computational and statistical challenges of traditional hierarchical Bayesian models with covariance (HBMc). The framework is evaluated against three central claims: (1) scalability through dramatically reduced computational cost, (2) precision under sparsity even with very few trials per block, and (3) improved interpretability via complementary information across resolution layers. Three Bayesian variants-Bayesian inference procedure (BIP; independent parameters), hierarchical Bayesian model with variance only (HBMv), and HBMc-were implemented in PyMC. The best-performing model, PyramidHBMc (HBMc at the top layer combined with HBMv refinement), consistently achieved the best performance (Watanabe-Akaike information criterion weight = 1.0), with the lowest root mean square error and standard deviation, reducing errors and variability by up to 74.1% and 78.5% relative to BIP across four datasets spanning one-dimensional temporal and two-dimensional spatial functions. These results directly support the three central claims and demonstrate the framework's broad applicability in perceptual and cognitive science.