When estimating regression coefficient via bayesian inference, don’t get point estimates, but instead posterior distributions from which we can obtain means, medians, quantiles, SD, credible intervals, etc. Why do we need/use Standard errors (se_mean)?
The Monte Carlo standard error (mcse) quantifies the uncertainty introduced by estimating expectations (e.g., means, quantiles, etc) with a finite number of MCMC draws. Was this what you were referring to?
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This is super confusing.
If you are doing frequentist inference, the standard error is the standard deviation of the estimator under resampling of the data. This plays the same role as the posterior standard deviation in Stan.
As @js592 pointed out, the se_mean reported by Stan is the standard error of the estimate of the mean using Monte Carlo draws. This is just the usual frequentist standard error, but it’s over Monte Carlo draws.
Technically, the estimate of se_mean is the estimate of posterior standard deviation divided by the square root of the effective sample size. This means that se_mean goes to zero as the number of draws goes to infinity. The posterior standard deviation will converge to a non-zero value and not move as the number of draws goes to infinity.
The critical thing to understand is that the se_mean will converge to
First, the standard error reported by Stan as se_mean is the frequentist standard error of estimating the mean with Monte Carlo draws. That is, it’s an estimate of the the standard deviation of the estimator under resampling.
With frequentist inference, the standard errors play the same role as the posterior standard deviation does in Bayesian inference.
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It can also be interpreted as the standard deviation of Bayesian posterior distribution of the expectation (where this posterior is due to finite number of Monte Carlo draws and different from the model parameter posterior). Instead of asking of what if we had observed some other Monte Carlo draws, we can ask what is our uncertainty about the expectation given we have observed those Monte Carlo draws we did actually observe. There is no practical difference, but if you want to stay full Bayesian, that is possible.
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Cool! I should be thinking of it this way rather than mechanically in terms of how we estimate it.
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The numerical computation is exactly the same, so you don’t need to change anything else than the language
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