🛠️ Bayes update on a grid

💡 Use Case

• Twenty lines that made 🎓 Lecture 14 - Updating beliefs with Bayes land: a prior, some data, a posterior you can see.
• I came back to that lecture on 21 October after Hamburg and a cold, two weeks behind, and understood the algebra of 📐 Bayes' rule without believing it. On the Wednesday I typed this out and watched a flat line turn into a hill.
• Since then it is the first thing I open when a question is how sure should we be after this little data. That question is the whole of 🔭 Bayesian thinking for product bets.

How I use it

1. Make a grid of 100 candidate rates between 0 and 1.
2. Give each a prior weight. Flat to begin with.
3. For each candidate, work out how likely the observed data would be if that rate were true.
4. Multiply prior by likelihood.
5. Divide by the total so the weights sum to one.
6. Plot prior and posterior on the same axes.

prior       ________________________   flat
data        7 successes in 10 tries
posterior   _______/‾‾‾\____           a hill around 0.7, and wide

⚖️ Pros & Cons

Pros

• No formula for the answer. It tries a hundred candidate values and weighs each one, which is what the rule says to do.
• Change the prior, run it again, and see how much the prior mattered. With ten observations: a lot. With five hundred: hardly.
• Short enough to retype from memory, which I did for the final.

Cons

• One unknown only. Two unknowns and the grid becomes a table, three and it becomes a bad idea.
• It makes a prior look easy to choose. Choosing it is the hard part and the script is silent on it.

🔄 Alternatives

• The closed-form update Hale showed on the board. Faster, and I did not trust it until the grid agreed with it.
• Drawing it by hand with ten candidate values. Rasmus Kjær did this in the margin and got there first.