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I just have 2 questions:

(1) If we can obtain samples from the posterior distribution, is there any need to try to compute posterior expectations and intervals analytically...?

(2) Also, I know that Monte Carlo methods will, in general, produce different results every time they are run, but how might I convince someone that my results obtained using MC are reliable?

Any help would be appreciated.

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1) Your question suggests you have the analytical posterior distribution available. If this is the case, it is always of value to have the analytic expectations and such like if it is non intractable to derive (as someone famous said, a good Monte Carlo is a dead Monte Carlo). For one thing, it may not be easy (algorithmically, or programmatically) to sample from the posterior distribution even if you know it analytically. However, if you don't know the analytical posterior expression i.e. you can't easily compute the integral denominator, then your only choice is to sample - that's why we have Monte Carlo in the first place.

2) If you produce multiple MC runs - you can compare results. If your MC sample can guarantee independent samples, then a simple variance estimator ($\frac{\widehat{\sigma}^2}{n}$ type) can give you some idea of how good your expectation estimate is. If your MC sample is from a MCMC algorithm, say, Metropolis Hastings, some more careful thought and good diagnostics are required - typically looking at the trace plots and autocorrelation plots. I'm sure someone else can contribute much more to this discussion...

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    $\begingroup$ +1 Although there is more that can be said, as you intimate, this is a nice start. (I don't want to leave you hanging with that cryptic remark, so I will be more explicit. One direction this conversation could go in would be to point out that any randomized sampling procedure is typically subject to greater uncertainties ("unreliability") than the output of a well-done MC calculation. This has a bearing on question (2).) $\endgroup$ – whuber May 2 '14 at 19:56
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    $\begingroup$ I guess the main thing is that the question is quite vague and open-ended. Perhaps Dan could consider being more specific? $\endgroup$ – queenbee May 3 '14 at 0:52
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For your second question, someone will be convinced if your MC results are converged within some error bar. Since multiple MC runs produce different results, the conception of converging result comes in. How to tell your results are converged or not is given in @queenbee answer.

By the way, for other people to fully reproduce your MC results, setting your random number seed at the beginning is important.

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