{"id":1593,"date":"2025-09-25T21:46:17","date_gmt":"2025-09-25T21:46:17","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=1593"},"modified":"2025-09-25T21:46:20","modified_gmt":"2025-09-25T21:46:20","slug":"the-poincare-problem","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2025\/09\/25\/the-poincare-problem\/","title":{"rendered":"The Poincar\u00e9 Problem"},"content":{"rendered":"\n

Selection bias<\/strong> is the hardest problem in statistics because it\u2019s almost unavoidable in practice, and once the data have been collected, it\u2019s usually not possible to quantify the effect of selection or recover an unbiased estimate of what you are trying to measure.<\/p>\n\n\n\n

And because the effect is systematic, not random, it doesn\u2019t help to collect more data. In fact, larger sample sizes make the problem worse, because they give the false impression of precision.<\/p>\n\n\n\n

But sometimes, if we are willing to make assumptions about the data generating process, we can use Bayesian methods<\/strong> to infer the effect of selection bias and produce an unbiased estimate.<\/p>\n\n\n\n

Click here to run this notebook on Colab<\/a>.<\/p>\n\n\n\n

Poincar\u00e9 and the Baker<\/h2>\n\n\n\n

As an example, let\u2019s solve an exercise from Chapter 7 of Think Bayes<\/em><\/a>. It\u2019s based on a fictional anecdote about the mathematician Henri Poincar\u00e9:<\/p>\n\n\n\n

\n

Supposedly Poincar\u00e9 suspected that his local bakery was selling loaves of bread that were lighter than the advertised weight of 1 kg, so every day for a year he bought a loaf of bread, brought it home and weighed it. At the end of the year, he plotted the distribution of his measurements and showed that it fit a normal distribution with mean 950 g and standard deviation 50 g. He brought this evidence to the bread police, who gave the baker a warning.<\/p>\n\n\n\n

For the next year, Poincar\u00e9 continued to weigh his bread every day. At the end of the year, he found that the average weight was 1000 g, just as it should be, but again he complained to the bread police, and this time they fined the baker.<\/p>\n\n\n\n

Why? Because the shape of the new distribution was asymmetric. Unlike the normal distribution, it was skewed to the right, which is consistent with the hypothesis that the baker was still making 950 g loaves, but deliberately giving Poincar\u00e9 the heavier ones.<\/p>\n\n\n\n

To see whether this anecdote is plausible, let\u2019s suppose that when the baker sees Poincar\u00e9 coming, he hefts k<\/code> loaves of bread and gives Poincar\u00e9 the heaviest one. How many loaves would the baker have to heft to make the average of the maximum 1000 g?<\/p>\n<\/blockquote>\n\n\n\n

How Many Loaves?<\/h2>\n\n\n\n

Here are distributions with the same underlying normal distribution and different values of k<\/code>.<\/p>\n\n\n\n

mu_true, sigma_true = 950, 50
<\/pre>\n\n\n\n
\"\"<\/figure>\n\n\n\n
As k<\/code> increases, the mean increases and the standard deviation decreases.<\/p>\n\n\n\n
When k=4<\/code>, the mean is close to 1000. So let\u2019s assume the baker hefted four loaves and gave the heaviest to Poincar\u00e9.<\/p>\n\n\n\n
At the end of one year, can we tell the difference between the following possibilities?<\/p>\n\n\n\n