Incorrect probability identity in Bayes estimator derivation

[duakhan1]

"Decision theoretic set estimation"
In the “Bayes estimator formalization” section, the step

Pr[X < c | Y] + Pr[X > d | Y] = Pr[X ≤ c | Y] − Pr[X ≤ d | Y]

is incorrect.

For a continuous posterior with CDF F:

Pr[X < c | Y] = F(c)
Pr[X > d | Y] = 1 − F(d)

So the correct expression is:

Pr[X < c | Y] + Pr[X > d | Y] = F(c) + 1 − F(d)

The derivation is missing the constant +1.

This does not affect the argmin, but the equality itself is mathematically incorrect and may confuse readers.

[duakhan1]

Hopefully this qualifies for the Bayesian workflow week's challenge!

[alexandrebouchard]

Thanks for the report!

If there would not be argmin, it would indeed be an error. But with the argmin the expression is mathematically correct.

But I agree that having an intermediate step in the derivation will make it easier to follow, so I will still grant it. Feel free to reference to this post in your submission.

[alexandrebouchard]

@junsongt