The framework of Solomonoff prediction assigns prior probability to
hypotheses inversely proportional to their Kolmogorov complexity. There are two
well-known problems. First, the Solomonoff prior is relative to a choice of
Universal Turing machine. Second, the Solomonoff prior is not computable.
However, there are responses to both problems. Different Solomonoff priors
converge with more and more data. Further, there are computable approximations
to the Solomonoff prior. I argue that there is a tension between these two
responses. This is because computable approximations to Solomonoff prediction
do not always converge.

Solomonoff 预测的框架将先验概率指派到假设上，这些假设与其 Kolmogorov 复杂度成反比。然而，Solomonoff 先验相对于通用图灵机的选择是有问题的，同时，Solomonoff 先验也是不可计算的。不过，对这两个问题有回应。同样的 Solomonoff 先验会随着越来越多的数据收敛。此外，对于 Solomonoff 先验也有可计算的近似值，但是作者认为在这两类问题的回应之间存在着紧张关系，因为可计算的 Solomonoff 预测近似并不总是能收敛。

Solomonoff 预测的困境

A Dilemma for Solomonoff Prediction

Algorithms for Magnetic Resonance (MR) image reconstruction from undersampled
measurements exploit prior information to compensate for missing k-space data.
Deep learning (DL) provides a powerful framework for extracting such
information from existing image datasets, through learning, and then using it
for reconstruction. Leveraging this, recent methods employed DL to learn
mappings from undersampled to fully sampled images using paired datasets,
including undersampled and corresponding fully sampled images, integrating
prior knowledge implicitly. In this article, we propose an alternative approach
that learns the probability distribution of fully sampled MR images using
unsupervised DL, specifically Variational Autoencoders (VAE), and use this as
an explicit prior term in reconstruction, completely decoupling the encoding
operation from the prior. The resulting reconstruction algorithm enjoys a
powerful image prior to compensate for missing k-space data without requiring
paired datasets for training nor being prone to associated sensitivities, such
as deviations in undersampling patterns used in training and test time or coil
settings. We evaluated the proposed method with T1 weighted images from a
publicly available dataset, multi-coil complex images acquired from healthy
volunteers (N=8) and images with white matter lesions. The proposed algorithm,
using the VAE prior, produced visually high quality reconstructions and
achieved low RMSE values, outperforming most of the alternative methods on the
same dataset. On multi-coil complex data, the algorithm yielded accurate
magnitude and phase reconstruction results. In the experiments on images with
white matter lesions, the method faithfully reconstructed the lesions.
Keywords: Reconstruction, MRI, prior probability, machine learning, deep
learning, unsupervised learning, density estimation

本文提出了一种利用无监督深度学习（Variational Autoencoders）学习全采样 MR 图像概率分布的显式先验术语的方法，从而弥补了 k 空间数据缺失的影响，而不需要训练成对的数据集。