Dice

Summary

Dice (: die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random values, commonly as part of tabletop games, including dice games, board games, role-playing games, and games of chance. A traditional die is a cube with each of its six faces marked with a different number of dots (pips) from one to six. When thrown or rolled, the die comes to rest showing a random integer from one to six on its upper surface, with each value being equally likely. Dice may also have polyhedral or irregular shapes, may have faces marked with numerals or symbols instead of pips and may have their numbers carved out from the material of the dice instead of marked on it. Loaded dice are designed to favor some results over others for cheating or entertainment. History Dice have been used since before recorded history, and it is uncertain where they originated. It is theorized that dice developed from the practice of fort

Official source

https://en.wikipedia.org/wiki/Dice

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We study the phenomenon of intransitivity in models of dice and voting. First, we follow a recent thread of research for n-sided dice with pairwise ordering induced by the probability, relative to 1/2, that a throw from one die is higher than the other. We build on a recent result of Polymath showing that three dice with i.i.d. faces drawn from the uniform distribution on {1,…,n} and conditioned on the average of faces equal to (n+1)/2 are intransitive with asymptotic probability 1/4. We show that if dice faces are drawn from a non-uniform continuous mean zero distribution conditioned on the average of faces equal to 0, then three dice are transitive with high probability. We also extend our results to stationary Gaussian dice, whose faces, for example, can be the fractional Brownian increments with Hurst index H∈(0,1). Second, we pose an analogous model in the context of Condorcet voting. We consider n voters who rank k alternatives independently and uniformly at random. The winner between each two alternatives is decided by a majority vote based on the preferences. We show that in this model, if all pairwise elections are close to tied, then the asymptotic probability of obtaining any tournament on the k alternatives is equal to 2−k(k−1)/2, which markedly differs from known results in the model without conditioning. We also explore the Condorcet voting model where methods other than simple majority are used for pairwise elections. We investigate some natural definitions of “close to tied” for general functions and exhibit an example where the distribution over tournaments is not uniform under those definitions.

2020

Deep Learning Building on Prior Ischemic Core Segmentation Improves Prediction of Infarction After Stroke

Elisabeth Anne Dirren, Guillaume Paul Olivier Leclerc, Dimitri Nestor Alice Van De Ville

Introduction: Imaging studies are used to guide patient selection for acute stroke treatment. Perfusion CT (pCT) is widely used to identify the acute ischemic core and penumbra, but the prediction of the final infarct remains challenging. With the advent of machine learning, algorithms learning the prediction of the final lesion from imaging data collected in the acute phase have been proposed. We aimed to investigate whether machine learning methods that integrate prior ischemic core segmentation improve the prediction of the final infarct after stroke. Methodology: We retrospectively included all stroke patients admitted to the Geneva University Hospital for intravenous and/or endovascular treatment from 01.2016 to 12.2017. All patients had acute pCT and follow-up MRI. An Attention-Gated 3D Unet was used as the baseline model on which the effect of access to a threshold-based ischemic core segmentation was tested. To ensure the efficient integration of information contained in voxels from the ischemic core, we extended the baseline model with a bayesian skip connection allowing only the prior to bypass most of the network. This modifies the model’s task to predict divergence from the prior representation. All models were evaluated for the prediction of the final infarct on follow-up MRI, given acute pCT maps as input. The output of each model was compared to finals lesions manually delineated by expert neurologists. Dice score was used to assess performance. Results: A total of 144 patients were included. Median hypoperfused tissue volume (Tmax > 6s) was 60 ml [17-134], median ischemic core (relative CBF < 38%) volume was 23 ml [17-33] and median final infarct volume was 13 ml [3-38]. Dice score for the threshold based ischemic core segmentation was 0.1. The baseline model with and without prior segmentation as input achieved a Dice score of 0.19. Adding the proposed bayesian skip connection lead to a more efficient integration of the prior segmentation ensuring faster convergence and better performance with a final Dice score of 0.21. Conclusion: The evaluated deep learning model can effectively leverage the information contained in a prior segmentation of the ischemic core to enhance the learning process and improve the prediction of the final infarct after stroke.

LIPPINCOTT WILLIAMS & WILKINS2021

Deep Learning Building on Prior Ischemic Core Segmentation Improves Prediction of Infarction After Stroke

Elisabeth Anne Dirren, Guillaume Paul Olivier Leclerc, Dimitri Nestor Alice Van De Ville

LIPPINCOTT WILLIAMS & WILKINS2021