英文标题：
《Bayesian Posteriors For Arbitrarily Rare Events》
---
作者：
Drew Fudenberg, Kevin He, and Lorens Imhof
---
最新提交年份：
2017
---
英文摘要：
We study how much data a Bayesian observer needs to correctly infer the relative likelihoods of two events when both events are arbitrarily rare. Each period, either a blue die or a red die is tossed. The two dice land on side $1$ with unknown probabilities $p_1$ and $q_1$, which can be arbitrarily low. Given a data-generating process where $p_1\\ge c q_1$, we are interested in how much data is required to guarantee that with high probability the observer\'s Bayesian posterior mean for $p_1$ exceeds $(1-\\delta)c$ times that for $q_1$. If the prior densities for the two dice are positive on the interior of the parameter space and behave like power functions at the boundary, then for every $\\epsilon>0,$ there exists a finite $N$ so that the observer obtains such an inference after $n$ periods with probability at least $1-\\epsilon$ whenever $np_1\\ge N$. The condition on $n$ and $p_1$ is the best possible. The result can fail if one of the prior densities converges to zero exponentially fast at the boundary.
---
中文摘要：
我们研究当两个事件任意罕见时，贝叶斯观察者需要多少数据才能正确推断两个事件的相对可能性。每一个周期，要么抛出一个蓝色骰子，要么抛出一个红色骰子。两个骰子落在1美元一边，概率未知，p\\u 1美元和q\\u 1美元，可以任意低。给定一个数据生成过程，其中$p\\u 1\\ge c q\\u 1$，我们感兴趣的是需要多少数据来保证观察者对$p\\u 1$的贝叶斯后验平均值很有可能超过$q\\u 1$的$（1-\\delta）c$倍。如果两个骰子的先验密度在参数空间内部为正，并且在边界处表现为幂函数，则对于每$\\ε>0，存在一个有限的$\\ N$，以便观察者在$\\ N$周期后获得这样的推断，每当$\\ np\\u 1\\ge N$时，概率至少为$\\ε$。n$和p\\u 1$的条件是最好的。如果一个先验密度在边界处以指数速度收敛到零，则结果可能失败。
---
分类信息：

一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
--
一级分类：Quantitative Finance 数量金融学
二级分类：Economics 经济学
分类描述：q-fin.EC is an alias for econ.GN. Economics, including micro and macro economics, international economics, theory of the firm, labor economics, and other economic topics outside finance
q-fin.ec是econ.gn的别名。经济学，包括微观和宏观经济学、国际经济学、企业理论、劳动经济学和其他金融以外的经济专题
--
一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
--

---
PDF下载：
-->