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Markov chain Monte Carlo
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==== Half-Width Test (Precision Check) ==== Once stationarity is accepted, the second part of the diagnostic checks whether the Monte Carlo estimator is accurate enough for practical use. Assuming the central limit theorem holds, the confidence interval for the mean <math>\mathbb{E}_\pi[g(X)]</math> is given by :<math> \bar{g}_n \pm t_{\alpha/2,\nu} \cdot \dfrac{\hat{\sigma}_n}{\sqrt{n}} </math> where <math>\hat{\sigma}^2</math> is an estimate of the variance of <math>g(X)</math>, <math>t_{\alpha/2,\nu}</math> is the [[Student's t-test|Student's <math>t</math>]] critical value at confidence level <math>1 - \alpha</math> and degrees of freedom <math>\nu</math>, <math>n</math> is the number of samples used. The '''half-width''' of this interval is defined as :<math> t_{\alpha/2,\nu} \cdot \dfrac{\hat{\sigma}_n}{\sqrt{n}} </math> If the half-width is smaller than a user-defined tolerance (e.g., 0.05), the chain is considered long enough to estimate the expectation reliably. Otherwise, the simulation should be extended.
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