DGI Mathematics

This page presents the complete mathematical framework for the Directional Grounding Index, including displacement vector analysis, the von Mises-Fisher distribution on the unit hypersphere, mean direction estimation, the concentration parameter \(\kappa\), and the linear normalization scheme.

Displacement Vectors

Given a question \(q\) and response \(r\), the displacement vector is:

\[ \boldsymbol{\delta} = \phi(r) - \phi(q) \in \mathbb{R}^n \]

This vector encodes the semantic transformation from question to response. Its two components carry distinct information:

• Direction \(\hat{\boldsymbol{\delta}} = \boldsymbol{\delta} / \|\boldsymbol{\delta}\|\): How the semantics changed (toward factual elaboration, toward tangential topics, toward contradictions, etc.)
• Magnitude \(\|\boldsymbol{\delta}\|\): How much the semantics changed (small for paraphrases, large for substantial elaboration)

DGI uses only the direction component, discarding magnitude. This is a deliberate design choice: the magnitude of semantic displacement varies widely across different question types and response lengths, but the direction of grounded displacement is consistent.

The Unit Displacement

The unit-normalized displacement vector is:

\[ \hat{\boldsymbol{\delta}} = \frac{\boldsymbol{\delta}}{\|\boldsymbol{\delta}\|} = \frac{\phi(r) - \phi(q)}{\|\phi(r) - \phi(q)\|} \]

This maps the displacement to the unit hypersphere \(S^{n-1} = \{\mathbf{v} \in \mathbb{R}^n : \|\mathbf{v}\| = 1\}\). All directional analysis --- the reference direction, the DGI score, the von Mises-Fisher model --- operates on \(S^{n-1}\).

Degenerate case

When \(\phi(r) = \phi(q)\) (the response is identical to the question in embedding space), \(\|\boldsymbol{\delta}\| = 0\) and the unit displacement is undefined. The implementation detects this case (\(\|\boldsymbol{\delta}\| < 10^{-8}\)) and returns DGI = 0, flagged = True.

The Reference Direction \(\hat{\boldsymbol{\mu}}\)

The reference direction is the mean direction of displacement vectors computed from a set of \(N\) verified grounded (question, response) pairs \(\{(q_i, r_i)\}_{i=1}^N\).

Computation

1. Compute each displacement: \(\boldsymbol{\delta}_i = \phi(r_i) - \phi(q_i)\)
2. Normalize to unit length: \(\hat{\boldsymbol{\delta}}_i = \boldsymbol{\delta}_i / \|\boldsymbol{\delta}_i\|\)
3. Compute the resultant vector: \(\mathbf{R} = \sum_{i=1}^N \hat{\boldsymbol{\delta}}_i\)
4. Normalize: \(\hat{\boldsymbol{\mu}} = \mathbf{R} / \|\mathbf{R}\|\)

This is equivalently expressed as:

\[ \hat{\boldsymbol{\mu}} = \text{normalize}\left(\frac{1}{N}\sum_{i=1}^{N} \hat{\boldsymbol{\delta}}_i\right) \]

Statistical Interpretation

The reference direction \(\hat{\boldsymbol{\mu}}\) is the maximum-likelihood estimate (MLE) of the mean direction parameter \(\boldsymbol{\mu}\) of a von Mises-Fisher distribution (see below). This makes it the optimal estimate under the assumption that grounded displacement directions are drawn from a unimodal directional distribution.

The Von Mises-Fisher Distribution

The von Mises-Fisher (vMF) distribution is the natural probability distribution for directional data on \(S^{n-1}\). It is the hyperspherical analog of the Gaussian distribution, and DGI uses it as the probabilistic model for grounded displacement directions.

Probability Density Function

The vMF distribution on \(S^{n-1}\) with mean direction \(\boldsymbol{\mu} \in S^{n-1}\) and concentration parameter \(\kappa \geq 0\) has density:

\[ f(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_n(\kappa) \exp\bigl(\kappa \, \boldsymbol{\mu}^\top \mathbf{x}\bigr) \]

where the normalizing constant is:

\[ C_n(\kappa) = \frac{\kappa^{n/2 - 1}}{(2\pi)^{n/2} I_{n/2 - 1}(\kappa)} \]

and \(I_\nu(\cdot)\) is the modified Bessel function of the first kind of order \(\nu\).

Interpretation of Parameters

Mean direction \(\boldsymbol{\mu}\): The mode of the distribution. This is the "most likely" direction. For DGI, \(\hat{\boldsymbol{\mu}}\) represents the characteristic displacement direction of grounded responses.

Concentration \(\kappa\): Controls how tightly the distribution is concentrated around \(\boldsymbol{\mu}\):

• \(\kappa = 0\): Uniform distribution on \(S^{n-1}\). No preferred direction.
• \(\kappa\) small: Spread out, weak directional preference.
• \(\kappa\) large: Tightly concentrated around \(\boldsymbol{\mu}\).
• \(\kappa \to \infty\): Point mass at \(\boldsymbol{\mu}\).

Why vMF for DGI?

The vMF distribution is the maximum-entropy distribution on \(S^{n-1}\) given a constraint on the mean direction. This makes it the least-informative assumption consistent with the observation that grounded displacements have a preferred direction. Using any other distributional family would impose additional structural assumptions not supported by the data.

Relationship to the Gaussian

In low dimensions, the vMF distribution is well-known:

• On \(S^1\) (the circle): vMF reduces to the von Mises distribution, the circular analog of the Gaussian.
• On \(S^2\) (the 2-sphere): vMF is the Fisher distribution, used extensively in paleomagnetic studies and geological directional data.

In high dimensions (\(n = 384\)), the vMF distribution concentrates rapidly for even moderate \(\kappa\) values, due to the vast surface area of \(S^{n-1}\).

Mean Direction Estimation (MLE)

Given \(N\) independent samples \(\hat{\boldsymbol{\delta}}_1, \ldots, \hat{\boldsymbol{\delta}}_N\) from \(\text{vMF}(\boldsymbol{\mu}, \kappa)\), the maximum-likelihood estimate of \(\boldsymbol{\mu}\) is:

\[ \hat{\boldsymbol{\mu}}_{\text{MLE}} = \frac{\mathbf{R}}{\|\mathbf{R}\|} = \frac{\sum_{i=1}^N \hat{\boldsymbol{\delta}}_i}{\left\|\sum_{i=1}^N \hat{\boldsymbol{\delta}}_i\right\|} \]

This is exactly what groundlens computes. The MLE for the mean direction does not depend on \(\kappa\) --- it is the same regardless of how concentrated the distribution is.

The Resultant Length \(\bar{R}\)

The mean resultant length is:

\[ \bar{R} = \frac{1}{N}\left\|\sum_{i=1}^N \hat{\boldsymbol{\delta}}_i\right\| = \frac{\|\mathbf{R}\|}{N} \]

This scalar in \([0, 1]\) measures the consistency of the directional data:

• \(\bar{R} \approx 0\): The unit vectors point in many different directions (no consistent pattern). The calibration data is noisy or from mixed domains.
• \(\bar{R} \approx 1\): All unit vectors point in nearly the same direction (strong consensus). The domain has a clear grounded displacement direction.

Concentration Parameter Estimation

The MLE for \(\kappa\) involves solving the equation:

\[ A_n(\kappa) = \bar{R} \]

where \(A_n(\kappa) = I_{n/2}(\kappa) / I_{n/2-1}(\kappa)\) is the ratio of Bessel functions. This equation has no closed-form solution.

Sra (2012) Approximation

groundlens uses the approximation from Sra (2012)¹ for the MLE of \(\kappa\):

\[ \hat{\kappa} \approx \frac{\bar{R}(n - \bar{R}^2)}{1 - \bar{R}^2} \]

This approximation is accurate for moderate to large \(n\) and avoids the computational cost of evaluating Bessel functions. For \(n = 384\) (our default embedding dimension), the approximation error is negligible.

Interpretation of \(\hat{\kappa}\)

\(\hat{\kappa}\)	Interpretation	DGI quality
> 10	High concentration --- strong directional consensus	Excellent discrimination
5--10	Moderate concentration	Good discrimination
1--5	Low concentration --- weak consensus	Poor discrimination
< 1	Near-uniform --- no meaningful direction	Calibration failure

Practical guidance

If \(\hat{\kappa} < 5\) after calibration, the reference direction is unreliable. Consider: (1) adding more calibration pairs, (2) filtering noisy pairs, (3) splitting into sub-domains if the data spans multiple topics.

The DGI Score

Given the reference direction \(\hat{\boldsymbol{\mu}}\) and a new (question, response) pair, the DGI score is:

\[ \text{DGI} = \hat{\boldsymbol{\delta}}^\top \hat{\boldsymbol{\mu}} = \cos\theta \]

where \(\theta\) is the angle between the displacement direction and the reference direction. This is the cosine similarity between the unit displacement and the unit reference direction.

Statistical Interpretation Under vMF

Under the null hypothesis that the displacement direction is drawn uniformly from \(S^{n-1}\) (i.e., the response is unrelated to the grounded pattern), the expected DGI score is:

\[ \mathbb{E}_{\text{uniform}}[\text{DGI}] = \mathbb{E}[\hat{\boldsymbol{\delta}}^\top \hat{\boldsymbol{\mu}}] = 0 \]

with variance \(\approx 1/n\). For \(n = 384\), the standard deviation is \(\approx 0.051\).

Under the alternative hypothesis that the displacement is drawn from \(\text{vMF}(\hat{\boldsymbol{\mu}}, \kappa)\):

\[ \mathbb{E}_{\text{grounded}}[\text{DGI}] = A_n(\kappa) = \frac{I_{n/2}(\kappa)}{I_{n/2-1}(\kappa)} \]

For well-calibrated domains (\(\kappa \geq 10\), \(n = 384\)), \(A_n(\kappa) \approx 1 - (n-1)/(2\kappa)\), which is close to 1.

The DGI threshold of 0.30 is approximately \(6\sigma\) above the null hypothesis mean, providing strong statistical confidence.

Linear Normalization

The DGI score lies in \([-1, 1]\) (the range of cosine similarity). The linear normalization maps this to \([0, 1]\):

\[ \text{DGI}_{\text{norm}} = \frac{\text{DGI} + 1}{2} \]

Why Linear (Not Tanh)?

Unlike SGI (which has a semi-infinite range requiring compression), DGI already lies in a bounded interval. A linear map preserves the metric structure:

• Equal differences in raw DGI correspond to equal differences in normalized DGI.
• The midpoint (DGI = 0, orthogonal to reference direction) maps to 0.5.
• The extremes (DGI = -1 and DGI = +1) map to 0 and 1 respectively.

A non-linear normalization (such as tanh) would distort the well-calibrated cosine similarity values and reduce the interpretability of the vMF concentration parameter.

DGI as a Sufficient Statistic

Under the vMF model, the DGI score \(\gamma = \hat{\boldsymbol{\delta}}^\top \hat{\boldsymbol{\mu}}\) is a sufficient statistic for the concentration parameter \(\kappa\). This means that the DGI score contains all the information that the full displacement direction \(\hat{\boldsymbol{\delta}}\) has about whether the response follows the grounded pattern. No additional information about grounding can be extracted from \(\hat{\boldsymbol{\delta}}\) beyond what is captured by DGI.

This is a consequence of the exponential family structure of the vMF distribution. The density can be written as:

\[ f(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_n(\kappa) \exp(\kappa \, \boldsymbol{\mu}^\top \mathbf{x}) \]

The sufficient statistic for \(\kappa\) is \(\boldsymbol{\mu}^\top \mathbf{x}\), which is exactly the DGI score (when \(\boldsymbol{\mu}\) is known or estimated).

Connection to Hypothesis Testing

DGI can be framed as a one-sided hypothesis test:

• \(H_0\): The displacement direction is uniform on \(S^{n-1}\) (no grounding signal).
• \(H_1\): The displacement direction follows \(\text{vMF}(\hat{\boldsymbol{\mu}}, \kappa)\) with \(\kappa > 0\) (grounded).

The DGI score is the test statistic. The threshold DGI = 0.30 defines the rejection region. Under \(H_0\), the probability of DGI > 0.30 is vanishingly small (since \(0.30 / 0.051 \approx 5.9\sigma\)), giving the test high specificity.

Under \(H_1\) with domain-specific calibration (\(\kappa \geq 10\)), the expected DGI for grounded responses is well above 0.30, giving the test high sensitivity. This explains the observed AUROC values of 0.90--0.99 with domain calibration.

Geometric Visualization

Consider the unit hypersphere \(S^{n-1}\) with the reference direction \(\hat{\boldsymbol{\mu}}\) at the "north pole." The DGI score is the cosine of the polar angle \(\theta\) from the north pole:

\[ \text{DGI} = \cos\theta \]

• Grounded responses cluster near the north pole (small \(\theta\), high DGI).
• Hallucinated responses are scattered away from the north pole (large \(\theta\), low DGI).
• The DGI = 0.30 threshold corresponds to \(\theta \approx 72.5°\).
• The DGI = 0.00 threshold corresponds to \(\theta = 90°\) (equator).

The vMF distribution gives the "density of grounded responses" as a function of polar angle: highest at the pole, decaying exponentially as \(\theta\) increases, with the rate of decay controlled by \(\kappa\).

Caching Strategy

Computing the reference direction requires embedding all calibration pairs, which involves \(2N\) forward passes through the sentence transformer. groundlens caches \(\hat{\boldsymbol{\mu}}\) by (model_name, reference_csv) key to avoid recomputation:

• First call with a given key: compute and cache.
• Subsequent calls with the same key: return cached result.
• Different CSV paths produce independent cache entries, allowing multiple domain calibrations to coexist.

References

• Marin, J. (2026). A Geometric Taxonomy of Hallucinations in LLMs. arXiv:2602.13224v3.
• Fisher, R. A. (1953). Dispersion on a sphere. Proceedings of the Royal Society A, 217, 295--305.
• Mardia, K. V., & Jupp, P. E. (2000). Directional Statistics. John Wiley & Sons.
• Sra, S. (2012). A short note on parameter approximation for von Mises-Fisher distributions. Computational Statistics, 27(1), 177--190.
• Banerjee, A. et al. (2005). Clustering on the unit hypersphere using von Mises-Fisher distributions. JMLR, 6, 1345--1382.

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1. Sra, S. (2012). A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of \(I_s(x)\). Computational Statistics, 27(1), 177--190. ↩