$$\mathcal{Z}_k(x)$$

Equation 6 · AI Agents & Systems

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

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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,

Meanings in this article

• $x$: the writing.

Equation guide → · Article →

Equation 10 · AI Agents & Systems

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

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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 $\mathcal{Z}_k(x)$ .

Meanings in this article

• $x$: the writing.

Equation guide → · Article →

Equation 3 · AI Agents & Systems

Retrieval Is an Evidence System, Not a Memory

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

The retrieval literature said the opposite. Lewis and colleagues introduced RAG explicitly as a combination of parametric and non-parametric memory, and the architectural point is that the second is a different kind of object, not an extension of the first [ 1 ] . Their formulation treats the retrieved passage as a latent variable to be marginalised over. Writing x for the query, y for the output, and $\mathcal{Z}_k(x)$ for the top k passages returned by a retriever with parameters $\eta$ :

Meanings in this article

• $x$: the writing.

Equation guide → · Article →

Equation 8 · AI Agents & Systems

Retrieval Is an Evidence System, Not a Memory

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Read the structure rather than the arithmetic. The generator is never conditioned on the corpus. It is conditioned on z — one span, or a handful — and its output distribution is a weighted account of what those spans support. Everything outside $\mathcal{Z}_k(x)$ has probability zero of influencing the answer, and the system has no representation of what it excluded. REALM made the same commitment on the training side, learning a latent retriever end-to-end so that the model attends over documents drawn from a corpus at pretraining, fine-tuning and inference time, and reporting 4–16% absolute gains on open-domain question answering over prior methods [ 2 ] .

Meanings in this article

• $x$: the writing.

Equation guide → · Article →

Equation 13 · AI Agents & Systems

Retrieval Is an Evidence System, Not a Memory

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Suppose retrieval succeeds. The relevant passage is in $\mathcal{Z}_k(x)$ , correctly chunked, in the prompt. The system may still fail, for a reason that has nothing to do with retrieval quality.

Equation guide → · Article →

Equation 17 · AI Agents & Systems

Retrieval Is an Evidence System, Not a Memory

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Because retrieval and generation are separable, an end-to-end score is nearly uninformative about which one broke. Decompose it. Let $\mathcal{Z}_k(x)$ be the retrieved set and S(x) the set of spans that would suffice to answer x :

Equation guide → · Article →

Equation 9 · AI Agents & Systems

Reliable AI Agents Are Control Systems, Not Chatbots

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

where the retriever assigns probability to documents z and the generator conditions on a selected set $\mathcal{Z}_k(x)$ [ 6 ] . The decomposition matters because retrieval and generation fail differently.

Equation guide → · Article →