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Bathyarchaeia represent a class of archaea common and abundant in sedimentary ecosystems. Here we report 56 metagenome-assembled genomes of Bathyarchaeia viruses identified in metagenomes from different environments. Gene sharing network and phylogenomic analyses led to the proposal of four virus families, including viruses of the realms Duplodnaviria and Adnaviria, and archaea-specific spindle-shaped viruses. Genomic analyses uncovered diverse CRISPR elements in these viruses. Viruses of the proposed family “Fuxiviridae” harbor an atypical Type IV-B CRISPR-Cas system and a Cas4 protein that might interfere with host immunity. Viruses of the family “Chiyouviridae” encode a Cas2-like endonuclease and two mini-CRISPR arrays, one with a repeat identical to that in the host CRISPR array, potentially allowing the virus to recruit the host CRISPR adaptation machinery to acquire spacers that could contribute to competition with other mobile genetic elements or to inhibit host defenses. These findings present an outline of the Bathyarchaeia virome and offer a glimpse into their counter-defense mechanisms.

We introduce a novel Information Criterion (IC), termed Learning under Singularity (LS), designed to enhance the functionality of the Widely Applicable Bayes Information Criterion (WBIC) and the Singular Bayesian Information Criterion (sBIC). LS is effective without regularity constraints and demonstrates stability. Watanabe defined a statistical model or a learning machine as regular if the mapping from a parameter to a probability distribution is one-to-one and its Fisher information matrix is positive definite. In contrast, models not meeting these conditions are termed singular. Over the past decade, several information criteria for singular cases have been proposed, including WBIC and sBIC. WBIC is applicable in non-regular scenarios but faces challenges with large sample sizes and redundant estimation of known learning coefficients. Conversely, sBIC is limited in its broader application due to its dependence on maximum likelihood estimates. LS addresses these limitations by enhancing the utility of both WBIC and sBIC. It incorporates the empirical loss from the Widely Applicable Information Criterion (WAIC) to represent the goodness of fit to the statistical model, along with a penalty term similar to that of sBIC. This approach offers a flexible and robust method for model selection, free from regularity constraints.