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

What does this equation mean?

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),

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Inputs and operationsF( X_1, ldots, X_n sim hatp_k )
Result or conditionhatp_k+1
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

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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p^k+1\hat{p}_{k+1}

Symbol hatp_k+1

the classical collapse setting.

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F\mathcal{F}

Symbol F

the fitting procedure.

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X1X_1

Symbol X_1

X1X_1 is an input to the expression that computes the quantity on the left.

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XnX_n

Symbol X_n

XnX_n is an input to the expression that computes the quantity on the left.

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p^k\hat{p}_k

Symbol hatp_k

the model fitted at generation k , and F\mathcal{F} the fitting procedure.

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=

=

The expressions on both sides represent the same quantity under the stated assumptions.

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addition

addition

Add the term after the plus sign to the term or group before it.

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subscript

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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How to interpret it

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What the article says around this equation

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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Sources cited in the article section

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