Parametric vs Kernelized Forms in Bias Modeling

• Parametric and kernelized forms are mathematical structures that define model behavior either via explicit low-dimensional parameterizations or implicit, nonparametric feature mappings.
• They influence bias by determining optimization landscapes, affecting phenomena such as tumor growth dynamics, tensor representations, and statistical learnability.
• Selecting the right form is critical for uncertainty quantification and bias mitigation, necessitating comprehensive sensitivity analyses in diverse applications from clinical therapy to machine learning.

Parametric and kernelized forms are central to the mathematical and statistical modeling of bias phenomena across domains ranging from medical therapy optimization, machine learning, combinatorics, LLMs, and social science. These forms precisely dictate how bias emerges, is measured, and ultimately can be mitigated or leveraged. The selection of a mathematical or algorithmic form—whether parametric (i.e., with explicit, low-dimensional parameterizations) or kernelized (i.e., where structure is encoded via nonparametric, often integral or feature-map-based constructions)—leads to sharply different phenomena and must be understood in both conceptual and operational detail.

1. Mathematical Specification and Governing Principles

Parametric and kernelized forms specify how model behavior depends on parameters and data input. In explicit, mechanistic models—such as those appearing in therapy optimization or tensor analysis—these forms are realized as governing equations or multilinear forms.

For example, in the design of cancer therapy schedules, parametric models for chemotherapy and radiotherapy formalize tumor cell dynamics under treatment via systems of ODEs with interpretable terms. Three canonical chemotherapy forms are as follows (Oh et al., 19 Nov 2025):

• Log-kill (LK): $\displaystyle\frac{dN}{dt} = rN - k(t)N$
• Norton–Simon (NS): $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$, $\delta<1$
• Eₘₐₓ (Saturable Kill): $$\displaystyle\frac{dN}{dt} = rN - \frac{E_{\max}k(t)}{EC_{50}+N}N$$

Each form directly encodes different biological/kinetic hypotheses, and the choice reduces or enriches the landscape of attainable model behaviors and optimal control policies, as articulated in Section 2 below.

In multilinear algebra and the study of Boolean functions, the parametric form is captured via tensor representations of multilinear maps (Bhrushundi et al., 2018):

• $d$-linear form: $f: (\mathbb{F}_2^k)^d \to \mathbb{F}_2$, represented as

$$f(X_1, \ldots, X_d) = \sum_{(i_1, \ldots, i_d) \in [k]^d} T(i_1, \ldots, i_d) X_{1,i_1}\cdots X_{d,i_d}$$

with parametric structure induced by tensor $T$.

Kernelized forms, by contrast, characterize structure via implicit feature mappings or convolution/integration, frequently used in bias audits and statistical learning but less common in explicit physical/biological modeling.

2. How Parametric Form Induces Bias in Optimization and Inference

The parametric or kernelized form of a model is not merely a technicality—it systematically biases the optimization landscape, affecting both the type and magnitude of "preferred" outputs.

Medical Scheduling (Chemotherapy/Radiotherapy)

Choice of ODE form fundamentally changes the optimal dose schedule—even under fixed total dose (Oh et al., 19 Nov 2025):

Model	Bias in Schedule Preference	Governing Reason
Log-kill	Back-loaded, late, large doses	Kill $\propto N$; dosing at large $N$ maximizes kill rate
NS	Intermediate, spread-out dosing	Growth term slows ($\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$0), balancing early/late dose
Eₘₐₓ	Fractionated, front-loaded dosing	Kill saturates at large $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$1 ($\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$2); early reduction is optimal

Equivalent patterns emerge for radiotherapy forms: linear–quadratic, proliferation–saturation, or time-window (CDR) models all prefer different sequencing and dosing, solely due to the model’s analytic form.

Multilinear Forms and Tensor Rank

In pseudorandomness and computational lower bounds, the parametric structure governs bias and its relation to learnability/correlation. For $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$3-linear forms over $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$4:

• Random $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$5-linear forms have exponentially small bias and thus extremely low correlation with low-degree polynomials (Bhrushundi et al., 2018).
• Small rank (low parametric complexity) induces large bias; i.e., decomposable forms (low-rank tensors) exhibit pronounced bias.

This establishes a strong correspondence: functional parametric form directly controls statistical or algorithmic bias.

3. Statistical and Algorithmic Implications

Model form constrains identifiability, uncertainty quantification, and the robustness of simulation-based inference.

• Model-dependent bias: Even when fitted to the same data, different parametric forms yield divergent (and sometimes contradictory) optimal predictions (Oh et al., 19 Nov 2025). Thus, recommendations may not generalize across model choices, necessitating full bias, sensitivity, and identifiability analyses rather than reliance on information criteria alone.
• Sensitivity to form: In the context of complex models (e.g., digital twins or adaptive personalized therapies), predictive bias may be much larger than parameter uncertainty when the analytic form is underdetermined.

Quantification frameworks thus demand:

• Evaluation of model form bias via endpoint outcomes (e.g., predicted tumor size, survival).
• Sensitivity analysis—varying both parameters and forms.
• Rigorous uncertainty quantification that encompasses model form bias, not just parameter or data noise.

4. Practical Methodologies for Bias Quantification and Mitigation

Addressing bias induced by parametric/kernelized forms requires methodical approaches:

(a) Comprehensive Model Comparison

Rather than optimizing within a single form, protocols must span all plausible model forms, quantifying:

• Form-dependent differences in optima.
• Sensitivity of recommendations to analytic assumptions (Oh et al., 19 Nov 2025).

(b) Empirical and Theoretical Boundaries

For multilinear/tensor models, bias bounds are available:

• The bias $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$6 for rank-$\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$7 $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$8-linear forms satisfies: $\displaystyle\frac{dN}{dt} = rN^{\delta} - k(t)N$9.
• Small bias (≈ pseudorandom functions) only possible for high-rank tensors, while low-rank imposes strong, learnable biases (Bhrushundi et al., 2018).

(c) Integrated Uncertainty and Identifiability

A modern uncertainty quantification workflow incorporates:

• Model selection based on form-bias analysis.
• Practical parameter identifiability: assessment of whether inferences are dominated by structural ambiguity versus statistical noise.
• Use of Bayesian or frequentist posterior predictive analyses that explicitly factor in form-induced variation (Oh et al., 19 Nov 2025).

5. Extensions and Broader Context

The centrality of parametric/kernelized form as the origin of bias is echoed in broader computational and statistical fields:

• In machine learning, different architectures (explicit parametric vs. kernelized) may bias learned representations in distinct, often subtle, ways. For example, in fairness-aware outlier detection, both the data distribution and the algorithm’s internal form (local vs. global, explicit vs. implicit) interact to produce or mitigate bias (Ding et al., 2024).
• In media bias detection, linguistic forms—parametric in their choice of grammatical structure, connotation, or frame—directly mediate measurable bias at every granularity from word to document (Spinde et al., 2023).
• In combinatorics and algebra, analytic form (e.g., sign of Fourier coefficients in modular forms) can be traced to root number or vanishing order in $\delta<1$0-functions, imposing characteristic sign biases on arithmetically meaningful objects (Koyama et al., 4 Sep 2025, Martin, 2016).

6. Comparative Analysis and Recommendations

The selection of a functional or kernelized form is not a neutral technical choice; it is a determinant of both the scope and magnitude of systematic bias in the mathematical, statistical, or algorithmic conclusions that follow.

• In high-stakes domains (e.g., clinical therapy optimization), the paramount recommendation is to benchmark against all plausible modeling forms, conducting explicit form-bias quantification and reporting sensitivity to form alongside parameter/posterior summaries (Oh et al., 19 Nov 2025).
• For analysis-focused domains (statistical learning, computational mathematics), parametric complexity (e.g., tensor rank, multilinearity) must be recognized as the governing factor for learnability, correlation resistance, and pseudorandomness (Bhrushundi et al., 2018).

A plausible implication is that advances in uncertainty-aware modeling and robust optimization will require frameworks where family-level model form bias—and not just parameter-level uncertainty—is first-class, explicitly quantified, and used as a basis for both selection and deployment.