this post was submitted on 17 May 2025
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The Collapse of GPT: Will future artificial intelligence systems perform increasingly poorly due to AI-generated material in their training data? (cacm.acm.org)
submitted 5 days ago by Pro@programming.dev to c/technology@lemmy.world
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[–] toastmeister@lemmy.ca 2 points 4 days ago

To make this more precise, we say that original data follows a normal distribution {\displaystyle X^{0}\sim {\mathcal {N}}(\mu ,\sigma ^{2})}, and we possess {\displaystyle M\_{0}} samples {\displaystyle X\_{j}^{0}} for {\displaystyle j\in {\\{\\,1,\dots ,M\_{0}\\,{}\\}}}. Denoting a general sample {\displaystyle X\_{j}^{i}} as sample {\displaystyle j\in {\\{\\,1,\dots ,M\_{i}\\,{}\\}}} at generation {\displaystyle i}, then the next generation model is estimated using the sample mean and variance:

{\displaystyle \mu \_{i+1}={\frac {1}{M\_{i}}}\sum \_{j}X\_{j}^{i};\quad \sigma \_{i+1}^{2}={\frac {1}{M\_{i}-1}}\sum \_{j}(X\_{j}^{i}-\mu \_{i+1})^{2}.}

Leading to a conditionally normal next generation model {\displaystyle X\_{j}^{i+1}|\mu \_{i+1},\\;\sigma \_{i+1}\sim {\mathcal {N}}(\mu \_{i+1},\sigma \_{i+1}^{2})}. In theory, this is enough to calculate the full distribution of {\displaystyle X\_{j}^{i}}. However, even after the first generation, the full distribution is no longer normal: It follows a variance-gamma distribution.