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Freshwater forcing from a retreating Antarctic Ice Sheet could have a wide range of impacts on future global climate. Here, we report on multi-century (present-2250) climate simulations performed using a fully coupled numerical model integrated under future greenhouse gas emissions scenarios IPCC RCP4.5 and 8.5, with meltwater discharge provided by a dynamic-thermodynamic ice sheet model. Accounting for Antarctica's meltwater contribution raises sub-surface ocean temperatures by 1 C at the ice margin, beyond rates currently projected. In contrast, 2-10 C cooler surface air and ocean temperatures in the Southern Ocean cause sea ice to expand and delay the increase of projected global mean anthropogenic warming. In addition, the projected loss of Arctic winter sea ice and weakening of the Atlantic Meridional Overturning Circulation are delayed be several decades. Our results demonstrate a clear need to more accurately account for meltwater input from ice sheets if we are to make confident climate predictions., Comment: 30 pages, 14 figures, submitted to Science Advances August 2019. This version is the original submission and does not reflect edits based on reviews. This work was also partially presented at AGU Fall Meeting in 2017 and 2018, AMS AOFD 2019, GCC 2019, AMS Annual 2020, AAG 2020

The Metropolis-Hastings method is often used to construct a Markov chain with a given $\pi$ as its stationary distribution. The method works even if $\pi$ is known only up to an intractable constant of proportionality. Polynomial time convergence results for such chains (rapid mixing) are hard to obtain for high dimensional probability models where the size of the state space potentially grows exponentially with the model dimension. In a Bayesian context, Yang, Wainwright, and Jordan (2016) (=YWJ) used the path method to prove rapid mixing for high dimensional linear models. This paper proposes a modification of the YWJ approach that simplifies the theoretical argument and improves the rate of convergence. The new approach is illustrated by an application to an exponentially weighted aggregation estimator.

Quantile regression is a tool for learning conditional distributions. In this paper we study quantile regression in the setting where a protected attribute is unavailable when fitting the model. This can lead to "unfair'' quantile estimators for which the effective quantiles are very different for the subpopulations defined by the protected attribute. We propose a procedure for adjusting the estimator on a heldout sample where the protected attribute is available. The main result of the paper is an empirical process analysis showing that the adjustment leads to a fair estimator for which the target quantiles are brought into balance, in a statistical sense that we call $\sqrt{n}$-fairness. We illustrate the ideas and adjustment procedure on a dataset of 200,000 live births, where the objective is to characterize the dependence of the birth weights of the babies on demographic attributes of the birth mother; the protected attribute is the mother's race.