CosmoMC: On the minimun \\Chi^2 and priors etc.
Posted: September 15 2005
Hi:
I am a beginner of MCMC and encounter several questions in fitting Dark Energy to current data as follows:
1) On the minimun \chi^2 , I have calculated it using action =0 and 2 in the params.ini of CosmoMC and set others as default . Using WMAP and SN data (Age and HST priors included) , parametrizing DE equation of state (EoS) as astro-ph/0407372 : w=w0+w1*(a-1)+w2*(a-1)^2 , I got the minimun \chi^2 /2= 805 and 810 respectively .Given MCMC cannot propose accurate \Chi^2 for best fit sample , the fact that the minimun \chi^2 /2 (action =0) being smaller than minimun \chi^2 /2 (action =2) is still puzzling for me . (I had supposed the former should be larger whatever)
2) On the prior , as for large EoS of DE in the early epoch it is unphysical since DE would dominate the universe .I hence used such a prior as :
When the total EOS of DE is larger than 0.05 today I will set the whole EOS as a constant 0.05 instead(this is somewhat optimistic but maybe applicable), my question is that would such a prior spoil the whole sampling of MCMC due to some unusual samplings around this point (w=0.05) and give an utterly different and wrong result? And which prior is more appropriate?
3) On the convergence level. Is it that for R-1<0.05 the (multi)chains must have converged and there are anyway no exceptions?
4)On the likelihood contour. There seems to be different claims on the likelihood contour of MCMC and I suppose two of them are very typical: one is astro-ph/0310723 where the way of using the contours out are exactly like(at least the 2-\sigma region in this way) those of marginalization with the conventional grids. The other is astro-ph/0406608 in page 5 of the eprint. The authors seemed to have used something like the coverage probability to work the 1-\sigma and 2\sigma regions out where the \delta\chi^2 is very strange and this often corresponds to some non-integral degrees of freedom. So what a method has COSMOMC used? The latter?
Thanks a lot.
Best,
Gong-Bo Zhao
I am a beginner of MCMC and encounter several questions in fitting Dark Energy to current data as follows:
1) On the minimun \chi^2 , I have calculated it using action =0 and 2 in the params.ini of CosmoMC and set others as default . Using WMAP and SN data (Age and HST priors included) , parametrizing DE equation of state (EoS) as astro-ph/0407372 : w=w0+w1*(a-1)+w2*(a-1)^2 , I got the minimun \chi^2 /2= 805 and 810 respectively .Given MCMC cannot propose accurate \Chi^2 for best fit sample , the fact that the minimun \chi^2 /2 (action =0) being smaller than minimun \chi^2 /2 (action =2) is still puzzling for me . (I had supposed the former should be larger whatever)
2) On the prior , as for large EoS of DE in the early epoch it is unphysical since DE would dominate the universe .I hence used such a prior as :
When the total EOS of DE is larger than 0.05 today I will set the whole EOS as a constant 0.05 instead(this is somewhat optimistic but maybe applicable), my question is that would such a prior spoil the whole sampling of MCMC due to some unusual samplings around this point (w=0.05) and give an utterly different and wrong result? And which prior is more appropriate?
3) On the convergence level. Is it that for R-1<0.05 the (multi)chains must have converged and there are anyway no exceptions?
4)On the likelihood contour. There seems to be different claims on the likelihood contour of MCMC and I suppose two of them are very typical: one is astro-ph/0310723 where the way of using the contours out are exactly like(at least the 2-\sigma region in this way) those of marginalization with the conventional grids. The other is astro-ph/0406608 in page 5 of the eprint. The authors seemed to have used something like the coverage probability to work the 1-\sigma and 2\sigma regions out where the \delta\chi^2 is very strange and this often corresponds to some non-integral degrees of freedom. So what a method has COSMOMC used? The latter?
Thanks a lot.
Best,
Gong-Bo Zhao