Can I use information about the distribution of the dependent to improve prediction? I'm trying to make predictions about a quantity on a per-subject basis.
If I aggregate my complete sample I can get very good fit for distributions like gamma or Weibull, so I can make some assumptions about how the dependent is distributed.
Can I use that information somehow to improve my predictions (like including a prior to my model)?
Ideally I would perform hierarchical modeling and include the subject as a random effect, but my dataset is too large to do something like this.
 A: In general, looking at just the response, $Y$, and trying to identify the corresponding distribution is meaningless. That's because when you model the response as a function of predictors, you assume that for each value of predictor, $x_i$,  the corresponding value of $Y_i$ comes from a certain distribution whose parameters are defined by $x_i$. Therefore the set of response values you have are generated not by one specific distribution, but by a mixture of distributions.
For instance, consider a simple model $Y = x + \epsilon$, $\epsilon \sim N(0, 1)$. Suppose my dataset has 100 observations for x = 10 and 200 observations for x = 150. If I create a histogram of $Y$ alone, I will see a bimodal distribution that is very different from Normal. However, saying that I should use some bimodal distribution to model $Y$ doesn't make sense. Instead, I should use Normal distribution whose mean depends on the covariate.
Doing what you did can make sense only if you have replicates. In my example you could create a histogram separately for x = 10 and x = 150, then you would see the correct picture: for a given value of the covariate, the response distribution is Normal.
