Bayesian vs Template Delensing

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Marius Millea, April 26, 2020

Summary

In this post, I examine the difference between no delensing, “MAP template delensing”, and Bayesian delensing. I find that at the noise levels of the 06b sims and if , Bayesian delensing offers improved constraints over template delensing. I believe this is because the Bayesian result actually accounts for the realization-dependent anisotropic and non-Gaussian statistics of the lensing residual noise.

In Carron 2018, Julien finds that at higher noise levels, there is no difference between template delensing and Bayesian delensing, and I reproduce this result too.

Simulated data used

I use my own flat-sky simulations on 650deg², 512² pixels at 3′ resolution (Bayesian delensing is not yet computationally feasible on the 06b sims themselves). No masking is included; I expect masking is orthogonal to what is explored here. I use exactly the 95GHz LAT 06b noise levels in conjunction with SAT 1/f noise specs.

  • noise ΔP = 0.68 μK-arcmin
  • beam FWHM = 2.2 arcmin
  • ℓknee = 60
  • αknee = 2.2

The simulated data has , since in this case we are most dominated by lensing residuals hence most sensitive to their statistics, which is what I want to test here.

Fig 1

The simulated data, just to get oriented:

Baylens4-maps.png

Delensing explored

No delensing

  • I use B data at ℓ<200 data only and compute the following posterior:

MAP template delensing

  • I compute the joint MAP lensed B template from E data and B data at ℓ>200. At the last S4 meeting I showed that in the absence of masking, this is identical to the marginal MAP lensed B template (which is what Julien has computed for the 06b sims).
  • I subtract the template from low-ℓ B data.
  • I then approximate the residual lensing as Gaussian isoptropic with a power-spectrum obtained from running the template estimation on many simulated data, yielding the following posterior:

Fig 2

The MC calculation of the residual lensing B modes for the template delensing method.

Baylens4-spectra.png

Bayesian delensing

  • I run a chain which samples the posterior .
  • This is the “right” answer with no approximations.
  • Note that this does not approximate the lensing residuals as Gaussian isoptropic, instead the chain correctly characterizes the extent to which the residuals are non-Gaussian and anisotropic, on a realization-dependent basis.

Results

Fig 3

The main result, constraints on from the methods compared. On this particular realization, Bayesian delensing achieves a 15% lower upper bound.

Baylens4-512 seed0.png

Fig 4

Same as above on four other realizations I ran.

Baylens4-512 seeds1234.png

Fig 5

Cross-correlation vs. the truth for the lensing estimate used to build the template, , and for the posterior mean from the chain. The fact that these are identical tells us the reason the chain does better is not because it has a better point estimate of .

Baylens4-phis.png

Comparison to Carron 2018

Julien also explored a very similar question in his paper Carron 2018. There are a few key differences between what I show here and what he did:

  1. He considered a higher noise level (ΔP = 2.3 vs. 0.68)
  2. He did not include 1/f noise
  3. He used a Laplace approximation to compute the integral .

There, he found no difference at all between MAP template delensing and his approximation of Bayesian delensing. When I use his noise levels, I reproduce his result exactly.

I suspect 2 & 3 are unimportant, but 1 makes it so that my result is more sensitive to the statistics of the lensing residual, as opposed to the Gaussian instrumental noise.

Fig 6

Same as above, but for noise levels from Carron 2018 (and only 100deg², which I don’t expect matters)

Baylens4-128 carron seed0.png