Equation 7 · Naive, Graph, and Agentic: A Systems Comparison of RAG Architectures

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p(y \mid x) \approx \sum_{z \in \mathcal{Z}_k(x)} p_\eta(z \mid x)\, p_\theta(y \mid x, z),

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This equation gives an approximation: it relates the quantities while allowing an approximation. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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p

Symbol p

p is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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y

Symbol y

y is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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x

Symbol x

the writing.

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z

Symbol z

z appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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\mathcal{Z}_k

Symbol Z_k

$Z_k$ appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

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p_\eta

Symbol p_eta

$p_e$ ta is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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p_\theta

Symbol p_θ

p_θ is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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≈

Approximately equal to; the equality is not exact.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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z \in \mathcal{Z}_k(x)

Starting index or lower bound: z in Z_k(x)

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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How to interpret it

Its accuracy depends on the assumptions and range of use described in the article.

What the article says around this equation

Start with the architecture that gives the pattern its name. A query is encoded, a retriever returns its top- k passages against a fixed index, and a generator conditions on those passages to produce one answer. No step revisits an earlier one. Lewis and colleagues’ original formulation makes the mechanism explicit: writing x for the query, y for the output, and z for a retrieved passage drawn from a top- k set $\mathcal{Z}_k(x)$ , the model marginalises over that set, $p(y \mid x) \approx \sum_{z \in \mathcal{Z}_k(x)} p_\eta(z \mid x)\, p_\theta(y \mid x, z)$ . with retriever parameters $\eta$ and generator parameters $\theta$ [ 1 ] . One retrieval call, one generation pass, and the entire system’s dependence on the corpus runs through that single set $\mathcal{Z}_k(x)$ …

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Start with the architecture that gives the pattern its name. A query is encoded, a retriever returns its top- k passages against a fixed index, and a generator conditions on those passages to produce one answer. No step revisits an earlier one. Lewis and colleagues’ original formulation makes the mechanism explicit: writing x for the query, y for the output, and z for a retrieved passage drawn from a top- k set $\mathcal{Z}_k(x)$ , the model marginalises over that set, $p(y \mid x) \approx \sum_{z \in \mathcal{Z}_k(x)} p_\eta(z \mid x)\, p_\theta(y \mid x, z)$ . with retriever parameters $\eta$ and generator parameters $\theta$ [ 1 ] . One retrieval call, one generation pass, and the entire system’s dependence on the corpus runs through that single set $\mathcal{Z}_k(x)$ .

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Sources cited in the surrounding passage

[1] Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks ↗

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