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Hi I am trying to estimate the posteriors of four calibration parameters namely $c_1, c_2, c_3$ and $c_4$ in the following equation using Bayesian inference

$$ F=c_1 \cdot (i^{c_2}) \cdot(s^{c_3}) \cdot (1-\exp(c_4 t)) $$

I have the observed data for the output $F$ and inputs $i$,$s$ and $t$. I know the range of $c_4$ from my prior knowledge, so I will use a uniform prior with this range for $c_4$. I don't have any prior knowledge about parameters $c_1, c_2, c_3$ individually. All I know is that $0 < c_1 \cdot (i^{c_2}) \cdot(s^{c_3}) < 1$ for all $i$ and $s$. Now I want to use Bayesian inference to find the posterior of parameters $c_1, c_2, c_3$ and $c_4$. Is there any way to use my indirect prior knowledge about $c_1, c_2, c_3$ here?

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  • $\begingroup$ How are you proposing to estimate the posteriors? $\endgroup$ – jbowman Aug 8 '17 at 1:16
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    $\begingroup$ If $i$ is an input, what does the constraint $0<c_1\,i^{c_2}\,c_3 < 1$ mean? For every $i$? (In $\mathbb{R}$? In some range? ) $\endgroup$ – Juho Kokkala Aug 8 '17 at 7:04
  • $\begingroup$ yeah, for every i. $\endgroup$ – MaMu Aug 8 '17 at 9:43
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    $\begingroup$ How does randomness come in? Is F a CDF or is F assumed to follow that equation with some random variation on top of what you observe? $\endgroup$ – Björn Aug 8 '17 at 9:52
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    $\begingroup$ Your model will not be identifiable with respect to $c_1$ and $c_3$, first merge the two parameters. $\endgroup$ – peuhp Aug 8 '17 at 10:07
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As a first try, I would suggest you building the multidimensional Jeffreys prior $\Pi(c_1,c_2,c_3,c_4)$ weighted by the support function returning 1 if your constraints are fullfilled and 0 else. This will give you a procedure to get a prior that will not depend from the way you choosed to parametrize $F$, which may be quite satisfying. I think it can be computed in a decent time. If some particular parameters of interest you can try to derivate the reference priors but this may be much more complicated.

An alternative approach if multiple fittings are performed and if it seems reasonable, would be to use hierarchical priors where each parameter have a prior whose parameters depends from other fittings (e.g. as in https://stats.stackexchange.com/a/245440/14346). Nevertheless adding the support condition on the $c_1, c_2, c_3$ may be cumbersome.

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  • $\begingroup$ Thanks for the suggestion. I would like to try multidimensional Jeffreys prior. But this is the first time I am using Bayesian inference and I don't have any knowldge on Jeffreys prior. Could you please provide a more detailed answer or direct me to somehwere where multidimensional Jeffreys prior has been applied in a similar porblem. Thanks. $\endgroup$ – MaMu Aug 8 '17 at 15:50

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