How to think about prior effects on posterior distributions

[emiruz]

@betanalpha what do you think about the prior fiddling ambitions in this example:

https://discourse.mc-stan.org/t/correlation-between-bias-and-true-parameter-values

Particularly:

@mrkm-a

I thought it might be because the prior is dragging the posterior towards 0, so I tested it again with a fixed true value (0.5) and the prior whose mean is the fixed value. As shown in the lower right panel, however, the bias still remains. An even more strongly negative correlation between bias and true values ( r = -0.65) is observed in the intercept parameter (figure not shown).

I am wondering if there is a way to figure out possible reasons for the correlation and bias, and how I can go about fixing them.