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

What does this equation mean?

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

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsprod_t=1^Tp(x_tmid x_<t)
Result or conditionp(x_1:T)
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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pp

Symbol p

p is part of the quantity the equation computes from the expression on the right.

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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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xtx_t

Symbol x_t

xtx_t is an input to the expression that computes the quantity on the left.

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x<tx_{<t}

Symbol x_<t

x_<t is an input to the expression that computes the quantity on the left.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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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

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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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Sources cited in the article section

These citations give research context. Read each source to check which claims it supports.

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