Equation 5 · From n-Grams to Reasoning Models: A Technical History of the Language Model
What does this equation mean?
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.
This equation states an equality: the expressions on both sides have the same value 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.
Read it piece by piece
Symbol c_i
is part of the quantity the equation computes from the expression on the right.
Symbol j
j occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol T_x
occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol alpha_ij
alphj is an input to the expression that computes the quantity on the left.
Symbol h_j
is an input to the expression that computes the quantity on the left.
Symbol e_ij
j occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol k
k occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol e_ik
k occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →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.
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.
See an illustrated explanation →Starting index or lower bound: j=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.
Ending index or upper bound: T_x
This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.
Denominator: sum_k=1^T_x exp(e_ik)
The complete quantity below the fraction bar; it must be nonzero for this division.
See an illustrated explanation →Starting index or lower bound: k=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.
Ending index or upper bound: T_x
This label says where the repeated addition, multiplication, or accumulation stops. It sets the last term or end of the range.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
Bahdanau, Cho and Bengio proposed letting the decoder search the source for the parts relevant to each output word, rather than reading from a single compressed vector [ 9 ] . The decoder computes, at each output step i , a context vector as a weighted sum of all encoder states: . The capacity of the intermediate representation now grows with the input rather than being fixed in advance. They further reported that the learned alignments corresponded well with human linguistic intuition — an interpretability result that arrived free with a performance fix, which is rare.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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