IndicParam: ML Benchmark and Parameter Identification

• IndicParam is a suite of innovations that spans ML evaluation on low-resource Indic languages, ODE parameter identifiability, and deterministic probabilistic inference.
• It leverages a human-curated multilingual benchmark and combines sensitivity analysis with profile likelihood methods to uncover minimal identifiable parameter combinations in ODE models.
• A deterministic, sampling-free approach using polynomial chaos expansion enables efficient Bayesian updates, offering significant computational savings in inverse problems.

IndicParam denotes a suite of distinct technical concepts developed within machine learning evaluation, parameter identifiability in ODE models, and probabilistic inverse problems. While each context introduces “IndicParam” as a major procedural or benchmarking innovation, the term is not uniform across fields. The following overview synthesizes these deployments with primary references to the relevant arXiv sources.

1. IndicParam as a Benchmark for Low-resource Indic Language LLM Evaluation

The “IndicParam” dataset is a human-curated benchmark for comprehensive evaluation of LLMs on low- and extremely low-resource Indic languages, as introduced by BharatGenAI (Maheshwari et al., 29 Nov 2025). Comprising 13,207 multiple-choice questions spanning 11 Indic languages (Nepali, Gujarati, Marathi, Odia as low-resource; Dogri, Maithili, Rajasthani, Sanskrit, Bodo, Santali, Konkani as extremely low-resource), plus a code-mixed Sanskrit–English subset, IndicParam targets the evaluation gap left by widespread reliance on high-resource language benchmarks.

Questions are mined from UGC-NET graduate-level examinations and are exhaustively OCR-processed, manually corrected, and annotated by native speakers. Each item is tagged by:

• Category: General Knowledge (GK) vs. Language Understanding (LU).
• Format: MCQ, assertion–reason, list matching, ordering, incorrect statement, fill-in-the-blanks.

No split into train/dev/test—IndicParam is strictly used for end-to-end evaluation. LLM performance is measured by overall and per-language accuracy, macro-averaged accuracy across languages, and category-specific breakdowns (GK vs LU).

In large-scale evaluation, state-of-the-art models (GPT-5, DeepSeek-V3.2, Claude-4.5, etc.) plateau below 50% overall accuracy (GPT-5: 45.0%), with notable degradation for extremely low-resource and morphologically complex languages (e.g., Bodo, Santali). LU questions show slightly higher success rates (GPT-5: 52.5%) but overall performance exposes systematic challenges in cross-lingual transfer, script diversity, and script-specific morphological complexity. The performance of mixture-of-expert (MoE) models is competitive on LU, but these models remain challenged on GK.

The benchmark is freely available, with associated scripts provided to standardize model evaluation.

Language	# Questions	LU %
Nepali	1,038	18.8
Gujarati	1,044	0.6
Marathi	1,245	4.7
Odia	577	20.3
...	...	...
Total	13,207

The statistical design, detailed annotation, and challenging question formats (e.g., assertion–reason, list-based matching, ordering) position IndicParam as a definitive benchmark to diagnose and advance the state of multilingual modeling for India’s linguistic landscape (Maheshwari et al., 29 Nov 2025).

2. IndicParam for Structural Identifiability in ODE Models

In the context of parameter identifiability, IndicParam refers to a systematic algorithmic pipeline for determining structurally identifiable parameter combinations in ODE models, as formulated by Eisenberg and Hayashi (Eisenberg et al., 2013). The core theoretical focus is on identifying which parameter combinations—rather than individual parameters—are uniquely recoverable from noise-free observations.

Theoretical Framework

Given an ODE model:

$$\dot{x}(t) = f(x, t, u, \theta), \quad y(t) = g(x, t, \theta)$$

structural identifiability is characterized by injectivity of the parameter-to-observation map $\Phi: \theta \mapsto y(\cdot; \theta)$. Lack of individual identifiability often implies the existence of functional dependencies:

$\varphi_i(\theta) = \text{constant}$

which are uniquely determined, even when single $\theta_j$ are not.

Algorithmic Steps

The IndicParam pipeline combines Fisher Information Matrix (FIM) subset selection and profile likelihood analysis to explicitly recover minimally identifiable parameter functions. The core strategy is:

1. Compute Sensitivity Matrix and FIM: Calculate sensitivities $\partial y/\partial \theta_i$ at a nominal parameter vector, assemble $F = X^T W X$.
2. Classify Parameters: Using the coefficient of variation (CV) from the covariance estimate $\sqrt{C_{ii}}/\theta^*_i$, identify trivially identifiable or insensitive parameters.
3. Subset Search: Identify “nearly full–rank” parameter subsets $A$ with $\mathrm{rank}\,F(A) = |A| - 1$ and $$\mathrm{rank}\,F(A \setminus \{\theta_i\}) = |A| - 1$$ for all $\Phi: \theta \mapsto y(\cdot; \theta)$0.
4. Profile Likelihood Analysis: For each such subset, fix $\Phi: \theta \mapsto y(\cdot; \theta)$1, re-estimate the remaining parameters, and fit pairwise parameter trajectories (typically rational curves).
5. Combination Recovery: Algebraically reconstruct the identifiable combination $\Phi: \theta \mapsto y(\cdot; \theta)$2 from the fitted parameter dependencies.
6. Model Re-parameterization and Experimental Design: Use recovered combinations to inform model simplification and to determine which further measurements would render all parameters identifiable.

Step	Description	Outcome
FIM analysis	Compute $\Phi: \theta \mapsto y(\cdot; \theta)$3 and its rank	Detect identifiability structure
Subset scan	Nearly full–rank search	Isolate minimal combinations
Profiling	Likelihood-based parameter trajectories	Functional dependencies (curves)
Algebra	Symbolic computation from fitted relations	Explicit parameter combinations

Examples in (Eisenberg et al., 2013) feature pharmacokinetics (2-compartment models), where minimal combinations such as $\Phi: \theta \mapsto y(\cdot; \theta)$4 or $\Phi: \theta \mapsto y(\cdot; \theta)$5 are recovered.

3. IndicParam for Deterministic Probabilistic Parameter Identification

In probabilistic inverse problems, “IndicParam” designates a deterministic, sampling-free method for parameter identification, as described by Jantsch, Webster, and Gunzburger (Rosić et al., 2012). This framework treats unknown parameters $\Phi: \theta \mapsto y(\cdot; \theta)$6 as random variables on a probability space, with inference phrased as a Bayesian update of the prior measure $\Phi: \theta \mapsto y(\cdot; \theta)$7 to a posterior $\Phi: \theta \mapsto y(\cdot; \theta)$8 given observation $\Phi: \theta \mapsto y(\cdot; \theta)$9.

Bayesian Formulation

Given likelihood $\varphi_i(\theta) = \text{constant}$0 and prior $\varphi_i(\theta) = \text{constant}$1, the posterior is:

$\varphi_i(\theta) = \text{constant}$2

Procedural Classification

• Measure-updating: Traditional MCMC or Kalman-type linear-Gaussian updates.
• Function-updating: Deterministic transformation (e.g. transport maps) updating the random variable rather than the measure.

IndicParam Deterministic Algorithm

The IndicParam paradigm builds a polynomial chaos expansion (PCE) surrogate for the forward map:

$\varphi_i(\theta) = \text{constant}$3

The Kalman update is performed directly in PCE coefficient space:

• Compute prior cross-covariance and covariance from PCE coefficients.
• Compute gain $\varphi_i(\theta) = \text{constant}$4.
• Apply update to zero- and first-order coefficients only; all subsequent steps are algebraic and deterministic.
• The posterior surrogate encodes updated uncertainty, allowing further inference or sampling as required.

This PCE-based approach eliminates the need for additional model solves after surrogate construction, yielding order-of-magnitude computational savings compared to MCMC, while maintaining accuracy even in nonlinear, non-Gaussian scenarios. Benchmarks include nonlinear diffusion and elasto-plastic parameters, where deterministic IndicParam demonstrates accuracy and credible variance estimates superior to ensemble Kalman filtering (Rosić et al., 2012).

4. Statistical and Computational Properties

In the LLM evaluation domain, IndicParam deploys macro-averaged, per-language, and per-category accuracy as principal metrics. Correlation analyses reveal that LLM performance on IndicParam scales reliably with model size for both GK and LU questions (Pearson $\varphi_i(\theta) = \text{constant}$5 for GK, $\varphi_i(\theta) = \text{constant}$6 for LU) (Maheshwari et al., 29 Nov 2025). Accuracy is sensitive to language resource tier and to question type, exposing current limitations in multilingual modeling.

In the ODE identifiability context, the critical property is the rank of the FIM—locally s-identifiable directions correspond to rank-deficiency, and nearly full–rank subsets encode minimal identifiable combinations (Eisenberg et al., 2013).

The deterministic probabilistic approach delivers computational complexity savings: for $\varphi_i(\theta) = \text{constant}$7 PCE basis functions, all updates post-surrogate require only $\varphi_i(\theta) = \text{constant}$8 operations, compared to $\varphi_i(\theta) = \text{constant}$9 for $\theta_j$0 MCMC samples (Rosić et al., 2012).

5. Applications, Implications, and Outlook

IndicParam as an LLM benchmark is actively used for revealing weaknesses in cross-lingual transfer and for motivating both dataset curation and architectural innovations in NLP for low-resource languages. The focus on diverse question types is especially valuable for discriminating between surface-level transfer and deeper linguistic competence.

In ODE modeling, IndicParam enables explicit computation of all structurally identifiable parameter combinations, guiding re-parameterization, experimental design, and targeted measurement. This leads to practical gains in model reduction and improved interpretability of inference pipelines.

For probabilistic parameter identification, IndicParam’s functional, sampling-free updates substantially accelerate realistic inverse problems, potentially underpinning real-time or large-scale applications where MCMC or ensemble methods are prohibitive.

A plausible implication is that further cross-adoption of these IndicParam innovations—spanning evaluation, identifiability theory, and deterministic inference—could inform new algorithmic paradigms within both machine learning and scientific computing.

• BharatGenAI et al., "IndicParam: Benchmark to evaluate LLMs on low-resource Indic Languages" (Maheshwari et al., 29 Nov 2025)
• Eisenberg & Hayashi, "Determining Structurally Identifiable Parameter Combinations Using Subset Profiling" (Eisenberg et al., 2013)
• Jantsch, Webster & Gunzburger, "Parameter Identification in a Probabilistic Setting" (Rosić et al., 2012)