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Published equation contexts

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

Why this formula appears here

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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x1:Tx_{1:T}

Symbol x_1:T

x1x_1:T is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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tt

Symbol t

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

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TT

Symbol T

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

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t=1t=1

Starting index or lower bound: t=1

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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TT

Ending index or upper bound: T

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

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How to interpret it

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

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

Equation 2 · History of Technology

From Autocomplete to Delegation: The Technical History Behind OpenAI Codex

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

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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