meeting 2024 02 12 n315 - JacobPilawa/TriaxSchwarzschild_wiki_5 GitHub Wiki

Context

  • In the last meeting, we were debating whether or not we should proceed with a round of rejection-sampled model points or points built from sampling inside a convex hull built from all model scalings. This bullet compares the resulting distribution of model points.

  • Background:

    • I've updated the convex hull calculation to do the convex hull fitting and sampling in (cos(theta), phi, T) space instead of (T, Tmaj, Tmin) space. Again, this fitting includes all of our scalings. This seems to have slightly improved the issue we were having near low Tmaj, but not to the same degree that the rejection sampling improves things, I believe.
    • I've built the rejection sampling by feeding in the set of best scaled models (~1500 models) with a cutoff K = 300 and nu = 1.5. The result is 328 models being used in the fit for the Gaussian Process.
      • One thing that I'm still working through -- part of our rejection sampling routine calculates the minimum of our Gaussian process chi2 value, and we base whether or not a model is "acceptable" based on this value + our cutoff. For some strange reason that I've yet to figure out, the Gaussian process model I've fit (using K = 300, nu = 1.5) says that the minimum chi2 value of our surface should be ~3288 instead of ~3456 which is our current best model. This meant that essentially all proposed points were getting rejected. If I force our rejection sampling routine to have the correct minimum for reference, things seem to work fine. I'm still trouble shooting this and trying to get this working without forcing it to have the proper minimum, but for now I think we're alright.

  • Takeaways:

    • The mass parameter distributions are very, very similar between the two methods with similar means and widths of the distribution.
    • With that sid, to me, it seems like the set of rejection sampled models better samples our current minimum/where we think our good fitting models should be for the shape/angle parameters. The updated convex hull sampling is still a bit offset from where our best model is, and that's not the case for the rejection sampled set of points. This is particularly bad for the edge of the space where we have a very small set of well-fitting points on one side of our minimum.

Diagnostic Plots

Convex Hulls

  • First, here's a look at the convex hulls built in the (cos(theta), phi, and T) space rather than the (T, Tmaj, Tmin) space. The panels below show the projections of the convex hull for cutoffs of 50, 150, 250, and 500. As a reminder, the convex hull "shrink-wraps" all the models within this cutoff and samples from inside this volume.
    • You can see that we still are in a similar position now as we were with (T, Tmaj, Tmin) space -- the current best fitting model in the (phi, cos(theta)) plane is extremely close to the upper right corner, and there are almost no good models further up-and-right, so the hulls don't "shrink wrap" this region unless we use a really large cutoff, which in turn skews the distribution of models to be not/centered/barely even include the best performing model!
Convex Hulls (50, 150, 250, 500 cutoffs)
images/240212/all_hulls_costheta-1.png

Rejection Sampling

  • I've had to use a rather generous cutoff of K = 300 for the rejection sampling routine to contain enough model points to be an accurate representation of the likelihood surface. Using the K = 300 cutoff results in about 325 models contained in the GPR fit. Here's the resulting posterior which we then use to reject models:
K = 300, nu = 1.5 posterior
images/240212/240205_gridAB_best_scalings_grid_alpha_K300_nu1.5-1.png

Resulting Model Points

  • The panels below show the set of best-scaled models, and the distribution of points for the convex hull and rejection sampling routines plotted as histograms. Both the convex hull and rejection sampled points contain 1000 models.

  • I included three choices of cutoff for the convex hull routine for comparison. I think the 150 case seems reasonable if we move forward with that, but again, I think I'd prefer to use the rejection sampled models.

  • Note that I've included both the (T, Tmaj, Tmin) and (theta, phi, and psi) panels for comparsion.

K=50 K=150 K=250
BH [images/240212/bh_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/bh_cut150.png) images/240212/bh_cut250.png
ML [images/240212/ml_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/ml_cut150.png) images/240212/ml_cut250.png
rho0 [images/240212/rho0_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/rho0_cut150.png) images/240212/rho0_cut250.png
T [images/240212/T_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/T_cut150.png) images/240212/T_cut250.png
Tmaj [images/240212/Tmaj_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/Tmaj_cut150.png) images/240212/Tmaj_cut250.png
Tmin [images/240212/Tmin_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/Tmin_cut150.png) images/240212/Tmin_cut250.png
theta [images/240212/theta_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/theta_cut150.png) images/240212/theta_cut250.png
phi [images/240212/phi_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/phi_cut150.png) images/240212/phi_cut250.png
psi [images/240212/psi_cut50.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/psi_cut150.png) images/240212/psi_cut250.png
  • And here are a set of identical posteriors (those used in building the rejection sampled points), with the proposed set of points plotted on top:
Rejection Sampled Points Convex Hull Sampled Points (K=150 case)
[images/240212/posteriors_with_rejection_points.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240212/posteriors_with_convex_points.png)