  Tweedie distributions are a special case of exponential dispersion models,
which are often used in classical statistics as distributions for generalized
linear models. Here, we reveal that Tweedie distributions also play key roles
in modern deep learning era, leading to a distribution independent
self-supervised image denoising formula without clean reference images.
Specifically, by combining with the recent Noise2Score self-supervised image
denoising approach and the saddle point approximation of Tweedie distribution,
we can provide a general closed-form denoising formula that can be used for
large classes of noise distributions without ever knowing the underlying noise
distribution. Similar to the original Noise2Score, the new approach is composed
of two successive steps: score matching using perturbed noisy images, followed
by a closed form image denoising formula via distribution-independent Tweedie's
formula. This also suggests a systematic algorithm to estimate the noise model
and noise parameters for a given noisy image data set. Through extensive
experiments, we demonstrate that the proposed method can accurately estimate
noise models and parameters, and provide the state-of-the-art self-supervised
image denoising performance in the benchmark dataset and real-world dataset.
