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Equation 7 · Part 10 · Reliable AI Agents Are Control Systems, Not Chatbots

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) = \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

Retrieval-augmented generation is routinely marketed as “giving the model your data.” The original RAG formulation was more precise: combine parametric memory in a sequence model with non-parametric memory in an external index. A simplified sequence-level expression is p(y∣x)=∑z∈Zk(x)pη(z∣x) pθ(y∣x,z)p(y \mid x) = \sum_{z \in \mathcal{Z}_k(x)} p_\eta(z \mid x)\,p_\theta(y \mid x,z). where the retriever assigns probability to documents z and the generator conditions on a selected set Zk(x)\mathcal{Z}_k(x) [ 6 ] . The decomposition matters because retrieval and generation fail differently.

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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.

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Sources cited in the surrounding passage

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