Hi elewer,
Sorry to answer so late, I hope this is still of interest to you!
I think you are right to be concerned. Mean-field ADVI cannot capture posterior correlations, and I think that multi-collinearity should give rise to correlations in the posterior, as you say.
What should end up happening, at least in a simple model (and yours doesn’t look too bad!), is that ADVI will get the posterior means approximately right, but it will likely underestimate the posterior variance, and it will completely fail to represent the posterior correlation. Taken together, this means that when you predict, you’re likely to get a good mean for the prediction, but a poor variance estimate.
If this is a concern, you could try full-rank ADVI, but I’m not sure I’d recommend it; it hasn’t worked that well for me in the past, and it can be hard to tell when it converges. May I ask why you’re considering ADVI in the first place – is NUTS too slow? If not, I’d go with that.
Hope this is helpful – let me know!
Best,
Martin