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        "description": "The method uses gradients with respect to continuous data, with differentiability, finite score expectations and vanishing boundary terms for integration by parts. Identifiability/correct specification and global optimization are separate requirements. Interior scores alone may fail to identify disconnected-support mixture weights. Pointwise laws of large numbers do not automatically establish a uniform argmin consistency theorem. Discrete/latent extensions and later diffusion methods remain separate.",
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      "statement": "# Score matching: learn a density model without its normalization constant\n\nHyvarinen introduced score matching: estimate a smooth continuous statistical model by matching data-gradients of log density, using integration by parts to obtain a sample objective that does not require its normalization constant.\n\nMany useful probability models can assign relative plausibility to data but cannot cheaply compute the constant that makes total probability equal one. Score matching offers another route to estimating their parameters. Differentiating the log density with respect to the data removes a normalization constant that does not depend on the data. Integration by parts then turns the unknown-data-density comparison into derivatives of the model evaluated at observed samples. The 2005 paper develops this objective and illustrates it with Gaussian models, independent component analysis and natural-image filters. Its Gaussian example recovers the usual sample mean and covariance when the required covariance inverse exists. This is a mathematical and statistical method designed by a human researcher, not an autonomous AI discovery. Smooth continuous support and boundary assumptions matter; finding a global optimum is not guaranteed. Discrete variables and intractable latent-variable models are explicitly left for future work. Later denoising-score methods and generative systems retain their own contributions.\n\n## Scientific scope\n\nThe method uses gradients with respect to continuous data, with differentiability, finite score expectations and vanishing boundary terms for integration by parts. Identifiability/correct specification and global optimization are separate requirements. Interior scores alone may fail to identify disconnected-support mixture weights. Pointwise laws of large numbers do not automatically establish a uniform argmin consistency theorem. Discrete/latent extensions and later diffusion methods remain separate.\n\n## People and systems\n\nAapo Hyvärinen\n\n\n\n## Original sources\n\n- [Hyvarinen, Estimation of Non-Normalized Statistical Models by Score Matching,JMLR6,695-709(2005)](https://jmlr.csail.mit.edu/papers/volume6/hyvarinen05a/hyvarinen05a.pdf)\n- [JMLR exact2005 score-matching publication identity](https://www.jmlr.org/beta/papers/v6/hyvarinen05a.html)\n",
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