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Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way. 

Herbarium collections shape our understanding of the world’s flora and are crucial for addressing global change and biodiversity conservation. The formation of such natural history collections, however, are not free from sociopolitical issues of immediate relevance. Despite increasing efforts addressing issues of representation and colonialism in natural history collections, herbaria have received comparatively less attention. While it has been noted that the majority of plant specimens are housed in the global North, the extent of this disparity has not been rigorously quantified to date. Here, by analyzing over 85 million specimen records and surveying herbaria across the globe, we assess the colonial legacy of botanical collections and how we may move towards a more inclusive future. We demonstrate that colonial exploitation has contributed to an inverse relationship between where plant biodiversity exists in nature and where it is housed in herbaria. Such disparities persist in herbaria across physical and digital realms despite overt colonialism having ended over half a century ago, suggesting ongoing digitization and decolonization efforts have yet to alleviate colonial-era discrepancies. We emphasize the need for acknowledging the inconvenient history of herbarium collections and the implementation of a more equitable, global paradigm for their collection, curation, and use.