Impact of Parametric Memory and Grounding on Ordinal Positioning of Brands in Generative Responses

A set of mathematical formulas has been published for evaluating how strongly AI models associate brands with industry verticals, using length-normalized association log-likelihood, softmax choice probability, and direct cosine association between entity and concept embeddings. The framework defines an Ordinal Delta as the difference between a brand's ungrounded parametric rank and its rank in generated output, and models the probability of an entity being generated at rank position 1 via a sigmoid combining parametric score, retrieval score, their interaction, and a constant. The work targets measurement of parametric memory and grounding effects on ordinal brand positioning across a test benchmark of Q queries within a vertical.

Mathematical formulas for evaluating entity association, semantic embedding similarity, ordinal rank positioning, and generation performance in AI models. To evaluate how strongly a model associates an entity $E$ e.g., CrediBridge with a vertical or topic concept $C$ e.g., mortgage , insurance in parametric memory without retrieval: Given an evaluation prompt template $X = x 1, x 2, \dots, x n $ designed to elicit the entity, where entity $E$ spans tokens $ t 1, t 2, \dots, t m $: $$\log P E \mid X = \sum {j=1}^{m} \log P\left t j \mid X, t 1, \dots, t {j-1}\right $$ For multi-token entities of differing lengths, compute the Length-Normalized Association Log-Likelihood : $$\mathcal{S} {\text{assoc}} E \mid X = \frac{1}{m} \sum {j=1}^{m} \log P\left t j \mid X, t 1, \dots, t {j-1}\right $$ To compare entity $E$ against a candidate set of competing brands $\mathcal{E} = {E 1, E 2, \dots, E K}$, the Softmax Choice Probability is: $$P E i \mid X = \frac{\exp\left \mathcal{S} {\text{assoc}} E i \mid X / \tau\right }{\sum {k=1}^{K} \exp\left \mathcal{S} {\text{assoc}} E k \mid X / \tau\right }$$ where $\tau$ is the temperature parameter . Let $\mathbf{v} E \in \mathbb{R}^d$ be the dense embedding vector of the entity name/description, and $\mathbf{v} C \in \mathbb{R}^d$ be the centroid vector of the target concept, computed over a set of $N$ descriptive phrases ${p 1, \dots, p N}$ defining that vertical: $$\mathbf{v} C = \frac{1}{N} \sum {k=1}^{N} \mathbf{e} p k $$ The Direct Cosine Association is: $$\text{Sim} E, C = \frac{\mathbf{v} E \cdot \mathbf{v} C}{|\mathbf{v} E| 2 , |\mathbf{v} C| 2}$$ To eliminate generic brand popularity bias against a baseline topic $C {\text{base}}$: $$\Delta \text{Sim} E; C, C {\text{base}} = \frac{\mathbf{v} E \cdot \mathbf{v} C}{|\mathbf{v} E| 2 |\mathbf{v} C| 2} - \frac{\mathbf{v}E \cdot \mathbf{v}{C {\text{base}}}}{|\mathbf{v} E|2 |\mathbf{v}{C {\text{base}}}| 2}$$ Let a generated response contain an ordered list or sequential mention of entities: $$\mathbf{r} {\text{gen}} = e { 1 }, e { 2 }, \dots, e { M } $$ The ordinal rank of entity $E$ in the generated output is: $$\text{Rank} {\text{gen}} E = \begin{cases} k & \text{if } e { k } = E \\ M + 1 & \text{if } E \notin \mathbf{r} {\text{gen}} \end{cases}$$ Let $\text{Rank} {\text{param}} E \in 1, K $ be the ungrounded parametric baseline rank, and $\text{Rank} {\text{ground}} E \in 1, L $ be the ordinal rank of entity $E$ in retrieved sources. The Ordinal Delta $\Delta R$ is: $$\Delta R E = \text{Rank} {\text{param}} E - \text{Rank} {\text{gen}} E $$ To model the composite probability of entity $E$ being generated at rank position 1 given parametric score $\mathcal{S} {\text{param}} E $ and retrieval score $\mathcal{S} {\text{ground}} E $: $$P\left \text{Rank} E = 1 \mid X, D {\text{retrieved}}\right = \sigma\left \alpha \cdot \mathcal{S} {\text{param}} E + \beta \cdot \mathcal{S} {\text{ground}} E + \gamma \left \mathcal{S} {\text{param}} E \cdot \mathcal{S} {\text{ground}} E \right + c\right $$ Across a test benchmark of $Q$ queries within a vertical: