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Equation 7 · Part 10 · Naive, Graph, and Agentic: A Systems Comparison of RAG Architectures

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

p(y∣x)≈∑z∈Zk(x)pη(z∣x) pθ(y∣x,z),p(y \mid x) \approx \sum_{z \in \mathcal{Z}_k(x)} p_\eta(z \mid x)\, p_\theta(y \mid x, z),
z∈Zk(x)z \in \mathcal{Z}_k(x)

What this part means

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.

Its job in the formula

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

The passage around this formula

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 Zk(x)\mathcal{Z}_k(x) , the model marginalises over that set, p(y∣x)≈∑z∈Zk(x)pη(z∣x) pθ(y∣x,z)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 η\eta and generator parameters θ\theta [ 1 ] . One retrieval call, one generation pass, and the entire system’s dependence on the corpus runs through that single set Zk(x)\mathcal{Z}_k(x)…

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Learn the underlying idea

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

Open the illustrated sums and products: repeat an operation over an index guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.