Equation 8 · From n-Grams to Reasoning Models: A Technical History of the Language Model
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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.
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Symbol θ
θ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Symbol E_x sim D, y sim pi_θ( × mid x)
sim D, y sim pi_θ( × mid x) appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.
Symbol r
r appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.
Symbol x
x appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.
Symbol y
y appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.
=
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.
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What the article says around this equation
The synthesis was to stop prompting for deliberation and start training for it, using the fact that some answers can be checked automatically. Lambert and colleagues named the technique in the open literature as Reinforcement Learning with Verifiable Rewards, applied within an otherwise conventional post-training pipeline [ 22 ] . The objective is unusually simple: . where r is not a learned preference model but a program: a unit test that passes, a numerical answer that matches, a proof that checks. Because r is exact, it cannot be gamed the way a learned reward model can, though it is only available where correctness is mechanically decidable.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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