99.44% pure

From the Guardian: Computer says there is a 80.58% probability painting is a real Renoir. The story goes on to say Dr Carina Popovici, Art Recognition’s CEO, believes that this ability to put a number on the degree of uncertainty is important.

It’s definitely valuable to put a number on the degree of uncertainty. What’s much less clear is that it’s valuable to put a number on the uncertainty to four-digit precision.   Let’s think about what it would take to be that precise.

If the 80.58% number was estimated from a proportion of observed data in some sense, quoting it to four digits would only make sense if the uncertainty was less than about 0.05%.  A standard error of 0.05% would need a sample size of more than five hundred million.

Another way you can get an estimate with high precision is including subjective expert opinion, which would be entirely appropriate in a context like this. There’s no limit to how precise this can be for the person whose opinion it is — you believe exactly what you believe — but there are very strong limits on how precise it can realistically be as a guide to others.  If the computer isn’t the one buying the Renoir, other people probably shouldn’t care about its opinion to more than one or two digits accuracy.

Sometimes when you come up with an estimate you want to quote it to higher precision than is directly useful — lots of statistical software, including some I write, quotes four or more digits in the default output. This allows rounding to happen closer to the point of use, such as before it’s in a headline in the mainstream media.