Do additional training points always decrease uncertainty?

[nbgl]

Say I have a predictor p trained on data points x₁, …, x_n. This predictor has a variance of v at t.

Say I also have a predictor p' trained on x₁, …, x_n+1. This has a variance of v' at t.

Is it necessarily true that v' ≤ v? My conjecture is yes, but I don’t seem to be able to prove it.

[nbgl]

I’m still curious about this. I wonder whether there’s a general result