[tfirst]

Holding all else equal, noisier estimates bias us towards the null. This is attenuation bias.

However, the estimates are still probably overestimated. Confounding, p-hacking, publication bias, all move us towards larger estimates.

[MichaelDickens]

I would think that, by default, noise would not have a bias? Adding noise doesn't change the mean, it just increases the variance, right?

[tfirst]

The Wikipedia page on this is not bad: https://en.wikipedia.org/wiki/Regression_dilution

[namibj]

It pushes confidence bounds closer to the null hypothesis.

[kingkawn]

“Still probably” is classic statistical science