Forged Invariant comments on A Bayesian Aggregation Paradox

[Forged Invariant]

Seeing the equations, it was hard to intuitively grasp why updates work this way. This example made things more intuitive for me:

If an event can have 3 outcomes, and we encounter strong evidence against outcomes B and C, then the update looks like this:

$$\underset{\text{Prior}}{\underset{}{\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)}}\times \underset{\text{Refute B}}{\underset{}{\left(\begin{array}{c}1\\ 0.01\\ 1\end{array}\right)}}\times \underset{\text{Refute C}}{\underset{}{\left(\begin{array}{c}1\\ 1\\ 0.01\end{array}\right)}}\ne \underset{\text{Pooled prior}}{\underset{}{\left(\begin{array}{c}1\\ 2\end{array}\right)}}\times \underset{\text{Refute B}}{\underset{}{\left(\begin{array}{c}1\\ 1.01\end{array}\right)}}\times \underset{\text{Refute C}}{\underset{}{\left(\begin{array}{c}1\\ 1.01\end{array}\right)}}$$

The information about what hypotheses are in the running is important, and pooling the updates can make the evidence look much weaker than it is.

[Jsevillamol]

Note that you are making the same mistake than me! Updates are not summarized in the same way as beliefs—for the update the “correct” way is to take an average of the $B,C$ likelihoods:

$\underset{\text{Posterior}}{\underset{}{\left(\begin{array}{c}1\\ 0.01\\ 0.01\end{array}\right)}}=\underset{\text{Prior}}{\underset{}{\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)}}\times \underset{\text{Refute B}}{\underset{}{\left(\begin{array}{c}1\\ 0.01\\ 1\end{array}\right)}}\times \underset{\text{Refute C}}{\underset{}{\left(\begin{array}{c}1\\ 1\\ 0.01\end{array}\right)}}\ne \underset{\text{Prior}}{\underset{}{\left(\begin{array}{c}1\\ 1+1\end{array}\right)}}\times \underset{\text{Refute B}}{\underset{}{\left(\begin{array}{c}1\\ \frac{0.01+1}{2}\end{array}\right)}}\times \underset{\text{Refute C}}{\underset{}{\left(\begin{array}{c}1\\ \frac{1+0.01}{2}\end{array}\right)}}\approx \underset{\text{Posterior}}{\underset{}{\left(\begin{array}{c}1\\ 0.5\end{array}\right)}}$

This does not invalidate the example though!

Thanks for suggesting, I think it helps clarify the conondrum.

[Forged Invariant]

The left hand side of the example is deliberately making the mistake described in your article, as a way to build intuition on why it is a mistake.

(Adding instead of averaging in the update summaries was an unintended mistake)

Thanks for explaining how to summarize updates, it took me a bit to see why averaging works.