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Equation 1 · Part 13 · From n-Grams to Reasoning Models: A Technical History of the Language Model

Starting index or lower bound: t=1

P(w1,w2,…,wT)=∏t=1TP(wt∣w1,…,wt−1).P(w_1, w_2, \ldots, w_T) = \prod_{t=1}^{T} P(w_t \mid w_1, \ldots, w_{t-1}).
t=1t=1

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

t=1 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

First, the object of study became prediction . A model of language is a probability assignment over what comes next. The chain rule makes this exact for any sequence of tokens: P(w1,w2,…,wT)=∏t=1TP(wt∣w1,…,wt−1)P(w_1, w_2, \ldots, w_T) = \prod_{t=1}^{T} P(w_t \mid w_1, \ldots, w_{t-1}). Second, quality became measurable without a task . Cross-entropy on held-out text is a number, and a lower number is unambiguously better. That gave the field a scalar to descend for the next seventy years, long before anyone knew what descending it would buy. Brown and colleagues later made the benchmark concrete, estimating an upper bound of 1.75 bits per character for English from a word trigram model measured against a balanced sample, and proposing a common corpus as a standard against which…

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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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