接种疫苗的人免受Covid-19影响的概率是多少？基于贝叶斯MCMC的已发表数据再分析，重点是应报告为“疗效”的内容
What is the probability that a vaccinated person is shielded from Covid-19? A Bayesian MCMC based reanalysis of published data with emphasis on what should be reported as 'efficacy'
摘要：
根据辉瑞、Moderna和阿斯利康在2020年底发布的新闻稿中传达的信息，我们建立了一个简单的贝叶斯模型，其中主要关注量在{ \em疫苗有效性}（“$ε$”）中起作用。由此产生的贝叶斯网络由马尔可夫链蒙特卡罗（MCMC）处理，在JAGS中实现，通过rjags与R接口。作为结果，我们得到了几个概率密度函数（pdf的）的$ε$，每个条件下的数据提供的三家制药公司。结果相对于参与试验的人数的大的变化是相当稳定的，并且在某种程度上与公司提供的结果很好地一致，在某种意义上，它们的值对应于MCMC产生的pdf的最可能值（“模式”），从而使我们确信我们的简单模型的有效性。然而，我们坚持认为，被报告为“疫苗效力”的数字应该是分布的平均值，而不是众数，因为拉普拉斯在大约250年前就已经非常清楚了（其“继承规则”来自最简单的问题）。在成功的数量等于试验数量的情况下，这一点尤其重要，因为Moderna声称对“严重形式”感染的疗效是100%。使用MCMC和近似公式，还显示了各种不确定性对接种疫苗的感染者的预测数量的影响。
英文摘要：
Based on the information communicated in press releases, and finally published towards the end of 2020 by Pfizer, Moderna and AstraZeneca, we have built up a simple Bayesian model, in which the main quantity of interest plays the role of {\em vaccine efficacy} (`$ε$'). The resulting Bayesian Network is processed by a Markov Chain Monte Carlo (MCMC), implemented in JAGS interfaced to R via rjags. As outcome, we get several probability density functions (pdf's) of $ε$, each conditioned on the data provided by the three pharma companies. The result is rather stable against large variations of the number of people participating in the trials and it is `somehow' in good agreement with the results provided by the companies, in the sense that their values correspond to the most probable value (`mode') of the pdf's resulting from MCMC, thus reassuring us about the validity of our simple model. However we maintain that the number to be reported as `vaccine efficacy' should be the mean of the distribution, rather than the mode, as it was already very clear to Laplace about 250 years ago (its `rule of succession' follows from the simplest problem of the kind). This is particularly important in the case in which the number of successes equals the numbers of trials, as it happens with the efficacy against `severe forms' of infection, claimed by Moderna to be 100%. The implication of the various uncertainties on the predicted number of vaccinated infectees is also shown, using both MCMC and approximated formulae.
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