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Equation 2 · Part 11 · From Autocomplete to Delegation: The Technical History Behind OpenAI Codex

Ending index or upper bound: T

p(x1:T)=∏t=1Tp(xt∣x<t).p(x_{1:T})=\prod_{t=1}^{T}p(x_t\mid x_{<t}).
TT

What this part means

This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.

Its job in the formula

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

The passage around this formula

An autoregressive model factorizes a sequence as p(x1:T)=∏t=1Tp(xt∣x<t)p(x_{1:T})=\prod_{t=1}^{T}p(x_t\mid x_{<t}). Source code can be placed in this representation alongside natural language. The model is not executing the program in this equation. It learns statistical structure over token sequences.

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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 →

See this notation across published equations →

Sources cited in the article section

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