m-murphy

... it says exactly what they are, q0 is an arbitrary density on R^d, and x is a realization from that density, p1 is the transformed density. This is intro graduate level probability.

Oh absolutely, so it's all the more important to be precise in what we're estimating and for what purpose, and to be honest about our ability to estimate it with appropriate uncertainty quantification (such as by using conformal prediction methods/bootstrapping).

Are the entities interacting in any way here? That's pretty critical for agent based models as they enable complicated/unexpected emergent behavior from simplified rules due to the interactions of the entities.

I think they mean either what is E[x| y] (standard regression point estimate) along with a confidence interval (this assumes that the mean is a meaningful quantity), or the interval s.t. F(x | y) -- the PDF of x -- is between .025 and .975 (the 95% predictive interval centered around .5). The point is that the width of the confidence interval around the point estimate of the mean converges to 0 as you add more data because you have more information to estimate this point estimate, while the predictive interval does not, it converges to the interval composed of the aleatoric uncertainty of the data generating distribution of x conditioned on the measured covariates y

Agreed! I also think it's extremely important as practitioners to know what we're even trying to estimate. Expected value (i.e. least squares regression) is the usual first thing to go for, does that even matter? We're probably actually interested in something like an upper quantile for planning purposes. And then the whole model component of it, the interval that's being simultaneously estimated is model driven and if that's wrong, then the interval is meaningless. There's a lot of space for super interesting and impactful work in this area IMO, once you (the practitioner) think more critically about the objective. And then don't even get me started on interventions and causal inference...

You're probably thinking of a predictive interval

That's not what a confidence interval is. A confidence interval is a random variable that covers the true value 95% of the time (assuming the model is correctly specified).