Context

• Cleaning up some of the work from last time now that the qualifying exam is out of the way.

• Last time, we wanted to run another set of models (~1100) in total to continue to fill out the space.

• Again, these models are chosen by:

  • Take the set (6478 in total) of best scaled models (from the original grid, where these chi2 are the NNLS chi2 values)
  • Cut these at K = 50 from the minimum. This leaves 4439 models.
  • Trim these by 1/4 by taking every 4th model.
    • We did this two times, resuling in ~2200 models)

• I've then re-run our GPR + dynesty routine on different combinations of the models to get a sense of how much stuff bounces around. Just as a reminder, we have:

  • The original, bad bins (only have access to the NNLS chi2)
  • The original, bad bins but rerun (where we have both kinematic and NNLS chi2)
  • The rebinned data (where we have both kinematic and NNLS chi2)

• Summary:

  • One thing that seems clear -- fixing the incorrect binning scheme seems to largely get rid of the bimodal shape that we were recoving. This could be important to note for diagnosing any problems in the future.
  • The NNLS/kinematic chi2 results from GPR and dynesty are largely in agreement with one another given a few caveats. First, it does seem like the chi2 and kinematic results differ slightly (but barely at the 1 sigma level), but are consistent with one another (i.e., the NNLS results are broadly consistent with one another, and the kinematic results are consistent with one another. The issue is really only when comparing the NNLS to the kinematic results). Given that things are individually consistent (in particular the two bad bin version NNLS results), I expect that, had we used the kinematic chi2 from the original grid, we would not find appreciable difference from the correct binning scheme.
  • All of this is to say that I think we're fine to move forward with our results posted below, coming from ~2200 models with the updated, correct binning scheme.
  • Perhaps the most important piece of news is that, after all of the changes and corrections of the bins, the parameters are not super significantly changed. The last set of plots on this page should compare those cases.
  • Old Params --> New Params:
    • BH: 5.5 (+1.0)(-0.8) --> 5.0 (+0.9)(-0.8)
    • M/L: 1.96 (+0.11)(-0.13) --> 1.96 (+0.13)(-0.14)
    • M15: 5.14 (+0.75)(-0.66) --> 4.09 (+0.55)(-0.48)
    • T: 0.31 (+0.06)(-0.05) --> 0.31 (+0.04)(-0.04)
    • Tmaj: 0.56 (+0.11)(-0.12) --> 0.74 (+0.10)(-0.15) : Note that we had imposed the prior on Tmaj to essentially be ~0.55 or so, hence the large discrepency in values and in errors.
    • Tmin: 0.03 (-0.02)(+0.05) --> 0.02 (+0.04)(-0.01)
    • u: 0.968 (+0.007)(-0.009) --> 0.957 (+0.01)(-0.01)
    • p: 0.940 (+0.010)(-0.013) --> 0.939 (+0.009)(-0.011)
    • q: 0.792 (+0.008)(-0.012) --> 0.784 (+0.009)(-0.010)

Plots

1D Panels for Example

• Note that the NNLS chi2s are largely in line with the K = 50 cutoff that we are using, but the kinematic chi2 bleeds out of the K = 50 region a bit. This is largely because we are running models based on the NNLS chi2 being within 50 of the minimum, rather than the kinematic chi2 being within that region. In practice this doesn't really impact anything since the number of models which bleed out of this region is quite small.

Original, Bad Bin NNLS	Rerun, Bad Bin NNLS	Rerun, Good Bin NNLS	Rerun, Bad Bin Kinem	Rerun, Good Bin Kinem	All Together
[images/240731/obn.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/rbn.png)	[images/240731/rgn.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/rbk.png)	[images/240731/rgk.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/all_together.png)

• And a comparsion of the chi2 distributions (normalized by total number of models since the grids differ by ~a few models due to taking longer on Expanse):

Plot
images/240731/CDF.png

Kinem and NNLS Comparions

• Here are two comparisons showing the kinem and NNLS chi2 from the rerun grids. The two track each other really quite well:

Kinematic Chi2 Comp	NNLS Chi2 Comp
[images/240731/kinem_comparison_240731.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/nnls_comparison_240731.png)

• And here are some comparisons of the original grid's NNLS chi2 vs the updated kinematic and NNLS chi2:
  • Seems like the NNLS chi2s are all tracking each other really quite well, but comparing the NNLS to kinematic chi2 is a bit more scattered. This is expected and what we were seeing last time.

Original, Bad Bid NNLS vs. Rerun Bad Bin NNLS	Original, Bad Bid NNLS vs. Rerun Good Bin NNLS	Original, Bad Bid NNLS vs. Rerun Bad Bin Kinem	Original, Bad Bid NNLS vs. Rerun Good Bin Kinem
[images/240731/OGnnls_originalnnls.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/OGnnls_rebinnednnls.png)	[images/240731/OGnnls_originalkinem.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/OGnnls_rebinnedkinem.png)

Cornerplots for Various Cases

• In each bunch of colors, the bottom point corresponds to K = 40, 50, 60, 80, 100 to the top, respectively.

K	Original, Bad Bin NNLS	Rerun, Bad Bin NNLS	Rerun, Good Bin NNLS	Rerun, Bad Bin Kinem	Rerun, Good Bin Kinem
40	[images/240731/OG_grid_alpha_K40_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K40_nu1.5-1.png)	[images/240731/rebinned_nnls_grid_alpha_K40_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K40_nu1.5-1.png)	images/240731/rebinned_grid_alpha_K40_nu1.5-1.png
50	[images/240731/OG_grid_alpha_K50_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K50_nu1.5-1.png)	[images/240731/rebinned_nnls_grid_alpha_K50_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K50_nu1.5-1.png)	images/240731/rebinned_grid_alpha_K50_nu1.5-1.png
60	[images/240731/OG_grid_alpha_K60_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K60_nu1.5-1.png)	[images/240731/rebinned_nnls_grid_alpha_K60_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K60_nu1.5-1.png)	images/240731/rebinned_grid_alpha_K60_nu1.5-1.png
80	[images/240731/OG_grid_alpha_K80_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K80_nu1.5-1.png)	[images/240731/rebinned_nnls_grid_alpha_K80_nu1.5-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K80_nu1.5-1.png)	images/240731/rebinned_grid_alpha_K80_nu1.5-1.png
100	[images/240731/OG_grid_alpha_K100_nu1.5-1.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K100_nu1.5-1.png)	[images/240731/rebinned_nnls_grid_alpha_K100_nu1.5-1.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K100_nu1.5-1.png)	images/240731/rebinned_grid_alpha_K100_nu1.5-1.png

Vertical Plot of the Cornerplots Above:

Plot
images/240731/vertical_comparison.png

Fancy Cornerplot Equivalents

• I've processed the cornerplots above in the same manner as we do our typical cornerplots. These would be our "final" posteriors.
• Note that I've processed all these below in the same way, including the original bad binning case. We had switched to imposing the prior on Tmaj

K	Original, Bad Bin NNLS	Rerun, Bad Bin NNLS	Rerun, Good Bin NNLS	Rerun, Bad Bin Kinem	Rerun, Good Bin Kinem
40	[images/240731/OG_grid_alpha_K40_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K40_nu1.5.d-1.png)	[images/240731/rebinned_nnls_grid_alpha_K40_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K40_nu1.5.d-1.png)	images/240731/rebinned_grid_alpha_K40_nu1.5.d-1.png
50	[images/240731/OG_grid_alpha_K50_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K50_nu1.5.d-1.png)	[images/240731/rebinned_nnls_grid_alpha_K50_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K50_nu1.5.d-1.png)	images/240731/rebinned_grid_alpha_K50_nu1.5.d-1.png
60	[images/240731/OG_grid_alpha_K60_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K60_nu1.5.d-1.png)	[images/240731/rebinned_nnls_grid_alpha_K60_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K60_nu1.5.d-1.png)	images/240731/rebinned_grid_alpha_K60_nu1.5.d-1.png
80	[images/240731/OG_grid_alpha_K80_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K80_nu1.5.d-1.png)	[images/240731/rebinned_nnls_grid_alpha_K80_nu1.5.d-1.png]] ](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K80_nu1.5.d-1.png)	images/240731/rebinned_grid_alpha_K80_nu1.5.d-1.png
100	[images/240731/OG_grid_alpha_K100_nu1.5.d-1.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_nnls_grid_alpha_K100_nu1.5.d-1.png)	[images/240731/rebinned_nnls_grid_alpha_K100_nu1.5.d-1.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/original_grid_alpha_K100_nu1.5.d-1.png)	images/240731/rebinned_grid_alpha_K100_nu1.5.d-1.png

A Final Direct Comparion

• Lastly, here are two cornerplots which show what's currently in the Overleaf vs. what the newest version of the parameters are. These are run with the same dynesty settings, with the difference being that we are not imposing a prior on Tmaj in the new version of the cornerplots.
• As a reminder, the old version was using the NNLS chi2, whereas the new version uses the kinematic chi2.

Current	Updated
[images/240731/old-1.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_5/wiki/[[images/240731/test_final_grid_alpha_K50_nu1.5.d-1.png)

Miscellaneous Plots