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Equation 11 · Part 12 · How Constitutional AI Actually Constrains a Model's Behavior

subscript

LPMAI(θ)=− E(x, yA, yB, p) ∼ DAI[p log⁡σ(rθ(x,yA)−rθ(x,yB))+(1−p) log⁡σ(rθ(x,yB)−rθ(x,yA))],\mathcal{L}^{AI}_{PM}(\theta) = -\,\mathbb{E}_{(x,\,y_A,\,y_B,\,p)\,\sim\, D_{AI}}\Big[p\,\log \sigma\big(r_\theta(x,y_A)-r_\theta(x,y_B)\big) + (1-p)\,\log \sigma\big(r_\theta(x,y_B)-r_\theta(x,y_A)\big)\Big],
subscript

What this part means

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.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

The passage around this formula

Nothing in that loss distinguishes where a pair came from; a human-labeled winner and an AI-labeled winner are interchangeable once written down as (x, ywy_w, yly_l) . That is the precise, narrow sense in which Constitutional AI “differs mechanically from plain RLHF”: it changes the labeling function for one half of one dataset, not the loss, not the optimizer, not the use of a KL penalty against the supervised policy. For the soft-labeled AI comparisons specifically, where the feedback model outputs a probability p of preferring response yAy_A over yBy_B rather than a hard choice, the corresponding cross-entropy term is LPMAI(θ)=− E(x, yA, yB, p) ∼ DAI[p log⁡σ(rθ(x,yA)−rθ(x,yB))+(1−p) log⁡σ(rθ(x,yB)−rθ(x,yA))]\mathcal{L}^{AI}_{PM}(\theta) = -\,\mathbb{E}_{(x,\,y_A,\,y_B,\,p)\,\sim\, D_{AI}}\Big[p\,\log \sigma\big(r_\theta(x,y_A)-r_\theta(x,y_B)\big) + (1-p)\,\log \sigma\big(r_\theta(x,y_B)-r_\theta(x,y_A)\big)\Big]. which exposes the other genuine mechanical difference:…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

Sources cited in the article section

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