On April 28th we need to submit the final version of the likelihood to the collaboration. I've been working on implementing changes and improvements to the likelihood. Here are the changes I've made thus far:

1. Changed luminosity function. I am currently using the Richards '06 luminosity function. Joe Hennawi thinks I should use a Hopkins, Richards, Hernquist (2007) combination function. I'm going to run on both and see which is better at targeting QSOs. See this post for more details.

2. Changed the redshift range of the QSO Catalog. Before we were only modeling QSOs from 2.0 < z < 3.9. Now we are modeling from 0.5 < z < 5.0. I need to optimize which range we should use for the numerator and which range we should use for the denominator. Nominally we are interested in targeting QSOs in the redshift range 2.15 < z < 3.5, so that is where I will start. Including low redshift QSOs in the denominator of the likelihood should improve the method significantly though.

3. Including BOSS QSOs along with SDSS DR5 QSOs with "good photometry." This fills out regions of the color/flux space which were clipped from SDSS such that we aren't missing QSOs which overlap with the horizontal branch stars. See this post for more details.

To speed up the computing of likelihoods, I am sub-sampling the low redshift QSOs (z > 2.1) by 10% and the high redshift (z < 2.1) by 50%. This reduces the number of objects in the QSO Catalog from 10,000,000 to 1,582,012. See this post for more details.

Joe Hennawi has some ideas for how to do this too which he outlined in an email today (see 4/20/2010 email subject: Likelihood weights):

Hi Guys,

After thinking about this more, and actually trying to work out the math, I convinced myself that indeed you were right. The factored probability I was proposing is not mathematically equivalent to the numerator which you are computing with the likelihood. Indeed, factoring the probability into p1(f_u/f_i/, f_g/f_i, f_r/f_i, f_z/f_i)*p2(f_i), is actually a more constrained model. This is however what we are doing with the extreme deconvolution, and for QSOs, this model is actually I think more desirable, for the following reason. Our model of QSO colors implicitly assumes that QSO colors do not depend on magnitude, and p1 reflects that behavior, whereas p2 is basically the magnitude prior. For stars, colors vary with magnitude. For extreme deconvolution, we had to fit to re-scaled fluxes (e.g. colors) rather than total fluxes in 5-d because gaussian models in general provide poor fits to power law distributions. This is also fine, since the color distributions of the stars vary so slowly with magnitude that you essentially make no errors by treating the probability as separable.

But this doesn't help too much with the current version of likelihood, since we don't plan to totally re-write that code. So I think the best approach would be to just take the entire list of QSOs that you plan to use, and re-scale them onto a grid of i-band fluxes spanning the range you want to consider, e.g. imin=17.8 -22.4. The grid spacing determines how finely you sample the i-band flux space, and this should be significantly smaller than the dispersion of QSO colors about the mean. Not sure what to choose here, but say you chose di = 0.01. Then your list of QSOs in the numerator would be a replication of that same list of QSOs N = (22.4-17.8)/0.01 = 460 times. Now in each i-magnitude bin, you can calculate the weights to apply to each QSO such that the weighted histogram of the QSO redshifts would give you the correct redshift and distribution and number of QSOs as predicted by the luminosity function.

I'm going to see how big of a difference it makes to calculate the likelihoods with my reduced catalog versus the full catalog (in terms of efficiency of selecting QSOs). Before I spend a bunch of time implementing Joe's suggestions.

1. Changed luminosity function. I am currently using the Richards '06 luminosity function. Joe Hennawi thinks I should use a Hopkins, Richards, Hernquist (2007) combination function. I'm going to run on both and see which is better at targeting QSOs. See this post for more details.

2. Changed the redshift range of the QSO Catalog. Before we were only modeling QSOs from 2.0 < z < 3.9. Now we are modeling from 0.5 < z < 5.0. I need to optimize which range we should use for the numerator and which range we should use for the denominator. Nominally we are interested in targeting QSOs in the redshift range 2.15 < z < 3.5, so that is where I will start. Including low redshift QSOs in the denominator of the likelihood should improve the method significantly though.

3. Including BOSS QSOs along with SDSS DR5 QSOs with "good photometry." This fills out regions of the color/flux space which were clipped from SDSS such that we aren't missing QSOs which overlap with the horizontal branch stars. See this post for more details.

To speed up the computing of likelihoods, I am sub-sampling the low redshift QSOs (z > 2.1) by 10% and the high redshift (z < 2.1) by 50%. This reduces the number of objects in the QSO Catalog from 10,000,000 to 1,582,012. See this post for more details.

Joe Hennawi has some ideas for how to do this too which he outlined in an email today (see 4/20/2010 email subject: Likelihood weights):

Hi Guys,

After thinking about this more, and actually trying to work out the math, I convinced myself that indeed you were right. The factored probability I was proposing is not mathematically equivalent to the numerator which you are computing with the likelihood. Indeed, factoring the probability into p1(f_u/f_i/, f_g/f_i, f_r/f_i, f_z/f_i)*p2(f_i), is actually a more constrained model. This is however what we are doing with the extreme deconvolution, and for QSOs, this model is actually I think more desirable, for the following reason. Our model of QSO colors implicitly assumes that QSO colors do not depend on magnitude, and p1 reflects that behavior, whereas p2 is basically the magnitude prior. For stars, colors vary with magnitude. For extreme deconvolution, we had to fit to re-scaled fluxes (e.g. colors) rather than total fluxes in 5-d because gaussian models in general provide poor fits to power law distributions. This is also fine, since the color distributions of the stars vary so slowly with magnitude that you essentially make no errors by treating the probability as separable.

But this doesn't help too much with the current version of likelihood, since we don't plan to totally re-write that code. So I think the best approach would be to just take the entire list of QSOs that you plan to use, and re-scale them onto a grid of i-band fluxes spanning the range you want to consider, e.g. imin=17.8 -22.4. The grid spacing determines how finely you sample the i-band flux space, and this should be significantly smaller than the dispersion of QSO colors about the mean. Not sure what to choose here, but say you chose di = 0.01. Then your list of QSOs in the numerator would be a replication of that same list of QSOs N = (22.4-17.8)/0.01 = 460 times. Now in each i-magnitude bin, you can calculate the weights to apply to each QSO such that the weighted histogram of the QSO redshifts would give you the correct redshift and distribution and number of QSOs as predicted by the luminosity function.

I'm going to see how big of a difference it makes to calculate the likelihoods with my reduced catalog versus the full catalog (in terms of efficiency of selecting QSOs). Before I spend a bunch of time implementing Joe's suggestions.

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