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I'm comparing different models and how well they fit but one of the models has two parameters $L_{0}$ and $\tau$ in the model:

$B+L_{0}e^{\frac{-t}{\tau}}$

My method of comparison for my single parameter models was just take the maximum posterior distribution but when I have two parameters how do I choose which to take? I know $0<L_{0}<10$ and $1<\tau<100$ and the maximum posterior can have most values of $L_{0}$ and $\tau$.

Hope that is making sense, ask if not, but how do you choose which values to take?

Lio Elbammalf
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  • Are you trying to choose between two Bayesian models, of which one has one parameter and the other has two parameters? – utobi Nov 13 '16 at 15:04

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As Utobi's comment suggests, it is not 100% clear what you are asking, but I suspect your problem is that you have an estimated model, and you want to do a residual analysis to check if your model fits OK to your data.

When I understand you correctly, you have so far used the highest point of a univariate posterior, calculated your predictions for this parameter, and checked the residual.

You CAN generalize this method to several parameters, by searching for the MAP, the point of Maximum Aposteriori Density. Important about the MAP is that you need to look for a multivariate density estimator - the MAP does in general NOT!!! conincide with the modes of the individual marginal distributions.

However, regardless of one or several parameters - using the "best" parameter for residual analysis is actually not the standard Bayesian way. Rather, the usual recipe is to draw many parameters from the posterior, calculate residuals for all of them, and calculate some statistics of your choice to see if the fitted models fit to the data. You will find examples of this under the keywords "Bayesian p-value" and "posterior predictive checks", see also here.

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Florian Hartig
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