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Equation 4 · Part 8 · Training on Your Own Output: Synthetic Data and What It Does to a Distribution

subscript

p^k+1=F(X1,…,Xn∼p^k),\hat{p}_{k+1} = \mathcal{F}\left( X_1, \ldots, X_n \sim \hat{p}_k \right),
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

Let p^k\hat{p}_k be the model fitted at generation k , and F\mathcal{F} the fitting procedure. The classical collapse setting is p^k+1=F(X1,…,Xn∼p^k)\hat{p}_{k+1} = \mathcal{F}\left( X_1, \ldots, X_n \sim \hat{p}_k \right). in which generation k+1 sees n samples drawn from its immediate predecessor and nothing else . The original data are gone. No filter selects among the samples. The lineage is single. Under those conditions, degradation compounds because there is no channel by which an error introduced at generation k can ever be corrected.

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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

These citations provide research context; check each source for the exact claim it supports.