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

When an AI answer ranks brands, two signals set where a brand lands: how strongly the model already associates it with the topic from training (parametric memory), and whether it appears in the sources retrieved at answer time (grounding). A brand with both is placed higher in the generated order than a brand carrying only one. The rest of this article defines each signal, measures the rank it produces, and sets out how to test whether the two together lift rank by more than each does on its own.

See: VISUAL EXPLAINER

1. Parametric Entity Association

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).

2. Semantic Embedding Association

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 \(C_{\text{base}}\):

$\Delta \text{Sim}(E; C, C_{\text{base}}) = \frac{\mathbf{v}_E \cdot \mathbf{v}_C}{|\mathbf{v}_E|_2 |\mathbf{v}_C|_2} - \frac{\mathbf{v}E \cdot \mathbf{v}{C_{\text{base}}}}{|\mathbf{v}_E|2 |\mathbf{v}{C_{\text{base}}}|_2}$

3. Ordinal Positioning & Rank Delta in RAG

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 \(\text{Rank}_{\text{par}}(E) \in [1, K]\) be the ungrounded parametric baseline rank, and \(\text{Rank}_{\text{gr}}(E) \in [1, L]\) be the ordinal rank of entity \(E\) in retrieved sources. The Ordinal Delta (\(\Delta R\)) is:

$\Delta R(E) = \text{Rank}_{\text{par}}(E) - \text{Rank}_{\text{gen}}(E)$

To model the composite probability of entity \(E\) being generated at rank position 1 given parametric score \(\mathcal{S}_{\text{par}}(E)\) and retrieval score \(\mathcal{S}_{\text{gr}}(E)\):

$P\left(\text{Rank}(E) = 1 \mid X, D_{\text{ret}}\right) = \sigma\left(\alpha \cdot \mathcal{S}_{\text{par}}(E) + \beta \cdot \mathcal{S}_{\text{gr}}(E) + \gamma \left(\mathcal{S}_{\text{par}}(E) \cdot \mathcal{S}_{\text{gr}}(E)\right) + c\right)$

4. Generation Rank Performance Metrics

Across a test benchmark of \(Q\) queries within a vertical:

1. Mean Reciprocal Rank (MRR): $\text{MRR}(E) = \frac{1}{|Q|} \sum_{q=1}^{|Q|} \frac{1}{\text{Rank}_{\text{gen}}^{(q)}(E)}$
2. Top-\(k\) Inclusion Rate (Recall@\(k\)): $\text{Inclusion}@k(E) = \frac{1}{|Q|} \sum_{q=1}^{|Q|} \mathbb{I}\left(\text{Rank}_{\text{gen}}^{(q)}(E) \le k\right)$

5. Testable Hypotheses

Sections 1 to 4 define the measurements. This section states the claim they are used to test: an entity that is present in parametric memory and in the retrieved sources is placed higher in the generated ordering than presence in either one alone would give it.

The sentinel rule in Section 3 covers only \(\text{Rank}_{\text{gen}}\). Extending it to the other two ranks makes \(\Delta R(E)\) defined for every entity in the candidate set \(\mathcal{E}\). Let \(\mathbf{r}_{\text{par}}\) be the ordered entities of the ungrounded response and \(\mathbf{r}_{\text{gr}}\) the entities of \(D_{\text{ret}}\) ordered by first mention, with the retrieved documents taken in retrieval order. An entity is in \(\mathbf{r}_{\text{gr}}\) if a retrieved document mentions it or is published by it.

$\text{Rank}_{\text{par}}(E) = \begin{cases} k & \text{if } e_{(k)} = E \text{ in } \mathbf{r}_{\text{par}} \\ K + 1 & \text{if } E \notin \mathbf{r}_{\text{par}} \end{cases}$

$\text{Rank}_{\text{gr}}(E) = \begin{cases} k & \text{if } e_{(k)} = E \text{ in } \mathbf{r}_{\text{gr}} \\ L + 1 & \text{if } E \notin \mathbf{r}_{\text{gr}} \end{cases}$

For query \(q\), the Parametric Set \(\mathcal{P}_q\) holds the entities in parametric memory and the Retrieval Set \(\mathcal{G}_q\) holds the entities in the retrieved sources:

$\mathcal{P}_q = {E \in \mathcal{E} : E \in \mathbf{r}^{(q)}_{\text{par}}}, \qquad \mathcal{G}_q = {E \in \mathcal{E} : E \in \mathbf{r}^{(q)}_{\text{gr}}}$

\(\Delta R\) in Section 3 measures movement relative to the parametric baseline. The Retrieval-Baseline Ordinal Delta measures movement relative to the retrieved sources:

$\Delta R_{\text{gr}}(E) = \text{Rank}_{\text{gr}}(E) - \text{Rank}_{\text{gen}}(E)$

A positive value in either delta means the entity was generated above its baseline position.

The claim splits into three hypotheses, each with the comparison group that isolates one effect:

1. Parametric Lift (H1): among entities in the retrieved sources, those also in parametric memory are generated further above their retrieval rank than those that are not: $\mathbb{E}\left[\Delta R^{(q)}_{\text{gr}}(E) \mid E \in \mathcal{P}_q \cap \mathcal{G}_q\right] > \mathbb{E}\left[\Delta R^{(q)}_{\text{gr}}(E) \mid E \in \mathcal{G}_q \setminus \mathcal{P}_q\right]$
2. Retrieval Lift (H2): among entities in parametric memory, those also in the retrieved sources are generated further above their parametric rank than those that are not: $\mathbb{E}\left[\Delta R^{(q)}(E) \mid E \in \mathcal{P}_q \cap \mathcal{G}_q\right] > \mathbb{E}\left[\Delta R^{(q)}(E) \mid E \in \mathcal{P}_q \setminus \mathcal{G}_q\right]$
3. Interaction (H3): in the rank-1 model of Section 3, the joint effect of the two scores exceeds the sum of their separate effects on the logit scale: $\gamma > 0$

The expectations run over the \(Q\) benchmark queries and the entities in each group. H1 and H2 are tested with a one-sided two-sample test on the group means. H3 is tested by fitting the Section 3 model by maximum likelihood on all \((q, E)\) pairs with outcome \(\mathbb{I}\left(\text{Rank}^{(q)}_{\text{gen}}(E) = 1\right)\), then testing \(H_0: \gamma = 0\) against \(\gamma > 0\).

Worked example. Four entities, so \(K = L = M = 4\). The ungrounded response orders them X, A, C, B. The retrieved sources order them A, B, C, X. The generated response orders them X, A, C, B.

The generated order equals the parametric order, so \(\Delta R = 0\) for every entity. \(\Delta R_{\text{gr}}\) shows X generated three positions above its retrieval rank, A one position below and B two below, which is the parametric order overriding the retrieval order.

Every entity here is in \(\mathcal{P}_q \cap \mathcal{G}_q\), so the comparison groups in H1 and H2 are empty for this query. When the generated and retrieved lists contain the same entities, \(\Delta R_{\text{gr}}\) sums to zero within a query (here \(3 - 1 + 0 - 2 = 0\)), so the group means carry information only when taken across the \(Q\) queries.