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Abstract How many copies of a fixed odd cycle, , can a planar graph contain? We answer this question asymptotically for and prove a bound which is tight up to a factor of 3/2 for all other values of . This extends the prior results of Cox and Martin and of Lv, Győri, He, Salia, Tompkins, and Zhu on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass on the edges of some clique maximizes the probability that edges sampled independently from form either a cycle or a path? 

Summary Distinguishing two models is a fundamental and practically important statistical problem. Error rate control is crucial to the testing logic, but in complex nonparametric settings can be difficult to achieve, especially when the stopping rule that determines the data collection process is not available. This paper proposes an $ e $-process construction based on the predictive recursion algorithm originally designed to recursively fit nonparametric mixture models. The resulting predictive recursion $ e $-process affords anytime-valid inference and is asymptotically efficient in the sense that its growth rate is first-order optimal relative to the predictive recursion’s mixture model. 

A central focus of data science is the transformation of empirical evidence into knowledge. As such, the key insights and scientific attitudes of deep thinkers like Fisher, Popper, and Tukey are expected to inspire exciting new advances in machine learning and artificial intelligence in years to come. Along these lines, the present paper advances a novel {\em typicality principle} which states, roughly, that if the observed data is sufficiently ``atypical'' in a certain sense relative to a posited theory, then that theory is unwarranted. This emphasis on typicality brings familiar but often overlooked background notions like model-checking to the inferential foreground. One instantiation of the typicality principle is in the context of parameter estimation, where we propose a new typicality-based regularization strategy that leans heavily on goodness-of-fit testing. The effectiveness of this new regularization strategy is illustrated in three non-trivial examples where ordinary maximum likelihood estimation fails miserably. We also demonstrate how the typicality principle fits within a bigger picture of reliable and efficient uncertainty quantification. 

Synopsis Cities, through the generation of urban heat islands, provide a venue for exploring contemporary convergent evolution to climatic warming. We quantified how repeatable the evolution of heat tolerance, cold tolerance, and body size was among diverse lineages in response to urban heat islands. Our study revealed significant shifts toward higher heat tolerance and diminished cold tolerance among urban populations. We further found that the magnitude of trait divergence was significantly and positively associated with the magnitude of the urban heat island, suggesting that temperature played a major role in the observed divergence in thermal tolerance. Despite these trends, the magnitude of trait responses lagged behind environmental warming. Heat tolerance responses exhibited a deficit of 0.84°C for every 1°C increase in warming, suggesting limits on adaptive evolution and consequent adaptational lags. Other moderators were predictive of greater divergence in heat tolerance, including lower baseline tolerance and greater divergence in body size. Although terrestrial species did not exhibit systematic shifts toward larger or smaller body size, aquatic species exhibited significant shifts toward smaller body size in urban habitats. Our study demonstrates how cities can be used to address long-standing questions in evolutionary biology regarding the repeatability of evolution. Importantly, this work also shows how cities can be used as forecasting tools by quantifying adaptational lags and by developing trait-based associations with responses to contemporary warming.