Inspired by Lê Nguyên Hoang's post on Bayesian Examination, I have been developing (as a hobby) a new app called Bayes-Up (available at: bayes-up.web.app). The app is now in a state where it is working well enough to be shared with others. In this post I list a few things you can do with it, because I expect that it will spark some interest within the community.

**Test and improve your calibration**: Bayes-Up uses a collection of good quality trivia questions from the open trivia database. The main point of the app is that you can find a list of multiple choice quizzes, answer questions by assigning probabilities to each of the possible choices, receive a score based off a quadratic proper scoring rule and later find statistics about the quality of your calibration. A good place to start is the quiz from the book*Factfulness*by Hans Rosling that I included in the app.**Create quizzes and upload them**. There exists already a small number of calibration training apps. Bayes-Up differs mainly because it allows to upload and share your own quizzes. This can solve one of the problems of calibration apps which is to create good quality content (quizzes / questions). If you are a teacher and want your students to develop more metacognitive skills and intellectual honesty, or if you are organizing workshops on probability calibration, Bayes-Up can make it easier for you. To add a quiz, simply write it in a spreadsheet, export it as a CSV file and upload it in Bayes-Up.**Recommend UI improvements, new features, report bugs, or contribute to the implementation**. Only very little feedback has been collected so far and certainly a lot could be improved with little effort. The code of the app is open source and hosted on github.**Analyse the data from Bayes-Up users**. So far about 30'000 questions have been answered by about 1'300 users since the end of December 2019. The collected data is available at this link and will likely grow in the following months. Simple questions that analysing this data could answer are: Do users become better calibrated over time? Is calibration topic-specific or transferrable? How can the answers of users with unknown calibration and unknown knowledge be aggregated to predict the right answers to every question? Let me know if you want to do something with it or need a better documentation.

Here you can see a graph of calibration of a user (available in the app):

https://twitter.com/le_science4all/status/1225498307348377600

And here you can see graphs of calibration for some of the quizzes of the app:

https://twitter.com/le_science4all/status/1225527782647705600

They clearly show overconfidence in the answers of the participants.

This is great!

One question is how are the error bars are calculated? From the description they are standard errors I think but if that's the case then you wouldn't really expect to get all of the black dots within the bars even if you were perfectly calibrated - more like 70% of dots within 1SE?

I'm also a bit confused by why, when I get 100% of questions correct when using a certain percentage, 100% isn't within my error bar?

You are right about the proportion of dots within the error bars. This sounds like something I would want to change.

100% is not within the error bar, because they are not exactly error bars, but bayesian estimations of where your true probability lies using a uniform prior between 0% and 100%. If I pick a coin which has a probability p of Head picked uniformly between 0% and 100%, then after observing 4 Heads out of 4 throws, you should still believe in average that the probability of Head is 80% ( = n_heads / (n_throws + 1) ) in average and a 75% confidence interval would not contain the probability 100%.

So you need to show more proofs that your 100% answers are indeed right 100% of the time. I agree this is confusing, and I want to change it for the better, but I am unsure how.

For all answers with probability p, I count the number of times it has been the right answer and a wrong answer. If anyone as a recommendation on how to compute the top and bottom percentage of the error bars from these, I would really appreciate it.

Thanks, I think I get it now.

If I observe 4 heads out of 4 and my prior was uniform across [0,1] then my posterior maximum likelihood is at 1 and this should definitely be within my error bars. Calculating the mean and adding symmetric error bars doesn’t work for asymmetric distributions.

To do this method more accurately you would have to calculate the full posterior distribution across [0,1] and use that to create error bars. Personally I would do this numerically but there may well be an analytical solution someone else will know about.

Alternatively, a frequentist approach: create error bars on the target percentage, rather than on the percentage achieved.

For each percentage grouping see how many questions had been answered using that percentage. Then use a binomial distribution to calculate the likelihood of each number of correct responses assuming that I am perfectly calibrated. This is essentially calculating a p-value with the null hypothesis being “I am perfectly calibrated”.

For example say I’ve answered 80% 4 times. If I’m perfectly calibrated I have a 0.8^4=41% chance of getting them all correct. Correspondingly I have:

0.8^3 x 0.2 x 4 = 41% to get 3 correct

0.8^2 x 0.2^2 x 6 = 15.4% to get 2 correct

0.8 x 0.2^3 x 4 = 2.5% to get 1 correct

0.2^4 = 0.2% to get 0 correct

If I am using a 90% CI (5% - 95%) then getting 0 correct is not inside my interval and nor is getting 1 correct (since 0.2% + 2.5% < 5%) but any of the other results are. So the top of my target error bar would reach to 100% and the bottom of would be between 25% and 50%

It is possible to combine all of the answers to create a single p-value across all percentages but this gets more complicated.

(Of course there would be 0 width error bars at 0% and 100% responses as any failures on these percentages are irrecoverable but this is right and proper)

Thanks for your recommendation! I have corrected the problem with the asymmetric distribution (now computing the whole distribution) and added a second graph showing exactly what you suggest and it looks good.

Unfortunately for the first approach that I implemented, the MAP is not always within a 90% confidence interval (It is outside of it when the MAP is 1 or 0). I agree that it is confusing and seems undesirable.

(You might need to hard-refresh the page if you want to see the update CTRL+SHIFT+R)

This looks great!

(I'm having a bug where the graph only displays results for 0-40% & 100% but I'm not sure if that's just my computer being weird)

It's because I changed it to only show estimations for probabilities which have received at least 4 answers and you have not yet answered enough questions. I am not confident that this change is good and I might revert it.

That's a surprisingly high number of people. Curious where they came from?

I promoted Bayes-up on my YouTube channel a couple of times 😋 (and on Twitter)

https://www.youtube.com/channel/UC0NCbj8CxzeCGIF6sODJ-7A/