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We propose a product-of-experts-type model utilizing Gaussian mixture experts and study configurations that admit an analytic expression for<jats:inline-formula><jats:alternatives><jats:tex-math>$$ f_Y (\\,\\cdot \\,, t) $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:msub><mml:mi>f<\/mml:mi><mml:mi>Y<\/mml:mi><\/mml:msub><mml:mrow><mml:mo>(<\/mml:mo><mml:mspace\/><mml:mo>\u00b7<\/mml:mo><mml:mspace\/><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In particular, with a focus on image processing, we derive conditions for models acting on filter, wavelet, and shearlet responses. Our construction naturally allows the model to be trained simultaneously over the entire diffusion horizon using empirical Bayes. We show numerical results for image denoising where our models are competitive while being tractable, interpretable, and having only a small number of learnable parameters. 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