# Change units in posterior MCMC Bayesian summary

I have 4000 iterations from an MCMC summarizing the posterior distribution. In the model used to estimate the posterior (a GLMM), my response variable was in g/m2, but I need to report the result in mg/m2. Can I just multiply each iteration of the posterior by 1000, or do I need to adjust the raw response unit and re-run the model? I assume the former is correct (just multiply), but I have not found direct examples of this approach on this site or in my Bayesian lit.

• You need more details. The answer to this question will depend on what GLMM you are using. Apr 2, 2018 at 19:59

Recall that if you define a new random variable as $Y = bX$, where $b>0$ is a constant, then it has probability density (or mass) function $f_Y(y) = f_X(y\,/\,b)\,/\,b$ and cumulative distribution function $F_Y(y) = F_X(y\,/\,b)$. Moreover, given the properties of expected value $E(bX) = b\, E(X)$ and given the properties of variance $\mathrm{Var}(bX) = b^2 \, \mathrm{Var}(X)$. Same goes about the median, since scaling does not change ordering of the values, so the "central point" after scaling, would be central point scaled. Per analogy, if mode was located at some point, then after scaling, the distribution would change it's shape, but the highest peak would just get scaled as everything else (since we just scaled the pdf/pmf!).
• Shouldn't the pdf be $f_Y(y)=f_X(y/b)/|b|$ ? Apr 2, 2018 at 20:34
• @VladislavsDovgalecs sorry, it's $b>0$, since it is a scale of the distribution, scale cannot be negative.