文章探讨了贝叶斯学派和频率学派在统计分析中产生不同结果的原因。两派都使用概率模型，但贝叶斯学派侧重于主观信念，而频率学派关注可重复事件。文章深入分析了双方在数据生成机制上的假设差异，以及何时停止信任主观判断的争论。频率学派倾向于保守，而贝叶斯学派则依赖先验信念。文章最后提出了一个值得科学研究的问题：在哪些情况下，个人的直觉判断比保守的极小化方法更有效？

🤔 贝叶斯学派和频率学派都使用概率模型，但对模型的应用有所不同。频率学派将概率视为可重复事件的建模工具，而贝叶斯学派用它来表示对任何陈述的主观信念程度。

💡 两派在解决统计问题时，对数据生成机制的假设存在差异。核心区别在于，何时停止信任我们对真实状态的信念。频率学派倾向于在某些时候放弃主观判断，避免使用反映个人信念的先验分布，而采取保守的策略。

🔍 频率学派认为，在设定分布参数后，不应依赖对真实状态的直觉，而应采取最坏情况的准备。这导致频率学派的置信区间通常比贝叶斯学派的置信区间更宽。

❓ 文章提出了一个关键问题：在什么情况下，个人的直觉判断比保守的极小化方法更有效？如果科学研究表明，在特定领域，平均科学家的判断优于极小化方法，那么频率学派可能会在该领域采取贝叶斯方法。

Published on July 3, 2025 5:47 PM GMT

Both schools use the same mathematical modeling tool, the probability measure, but they use it to model different things in the real world. Frequentists use it as a model for repeatable events like tossing a coin, while Bayesians use it to represent the degree of subjective belief to any statement.

I am more interested in the main causes of why Bayesian and frequentist statistical analysis gives different results on the same problem and the same data. Both camps, while trying to solve a given statistical problem, often make certain assumptions about the mechanism that generates the data. The core distinction and the source of fierce debate between them seems to be deciding when should we stop trusting our beliefs.

Frequentists recognize the importance of Bayes procedures as well. Wald proved that under mild conditions the set of Bayes procedures (with all possible priors) is complete. This means that for any non-Bayes procedure, you can find a Bayes procedure whose expected risk is never more, whatever the true state of nature is. Wald also proved that under the same mild conditions any minimax procedure is equivalent to some Bayes procedure with a "least favorable prior". From a Bayesian perspective, such a prior represents believing with certainty the most unfortunate state of nature.

Frequentists argue that after some point (usually setting the parameters of a family of distributions) we should not trust our intuitions about the true state of nature, so we should not choose a prior distribution which reflects our beliefs. Instead, we should behave like a paranoiac who's prepared for the worst case scenario. That is why, it seems, a frequentist %95 confidence interval is almost always wider than a Bayesian %95 credence interval for example.

The resolution of this debate seems to require a scientific investigation of the following question: Under which circumstances and for which problems a person's intuitive judgements are more useful than a paranoiac minimax approach? 

If an average scientist's judgements can be shown to be superior to the minimax approach in a given domain (using frequentist techniques :) then frequentists, I believe, will be convinced to behave like a Bayesian in such domains as well. I don't know if there is any literature on this kind of a question. 

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