Given a default distribution$P$and a set of test data$x^{M}=\{x_{1},\ x_{2},\ \ldots,\ x_{M}\}$this paper seeks to answer the question if it was likely that$x^{M}$was generated by$P$. For discrete distributions, the definitive answer is in principle given by Kolmogorov-Martin-Lof randomness. In this paper we seek to generalize this to continuous distributions. We consider a set of statistics$T_{1}(x^{M}), T_{2}(x^{M}),\cdots$. To each statistic we associate its maximum entropy distribution and with this a universal source coder. The maximum entropy distributions are subsequently combined to give a total codelength, which is compared with$-\log P(x^{M})$. We show that this approach satisfied a number of theoretical properties. For real world data$P$usually is unknown. We transform data into a standard distribution in the latent space using a bidirectional generate network and use maximum entropy coding there. We compare the resulting method to other methods that also used generative neural networks to detect anomalies. In most cases, our results show better performance.

This paper develops bounds for learning lossless source coding under the PAC (probably approximately correct) framework. The paper considers iid sources with online learning: first the coder learns the data structure from training sequences. When presented with a test sequence for compression, it continues to learn from/adapt to the test sequence. The results show, not unsurprisingly, that there is little gain from online learning when the training sequence length is much longer than the test sequence length. But if the test sequence length is longer than the training sequence, there is a significant gain. Coders for online learning has a somewhat surprising structure: the training sequence is used to estimate a confidence interval for the distribution, and the coding distribution is found through a prior distribution over this interval.