Bayesian vs Template Delensing
Marius Millea, April 26, 2020
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.
The simulated data, just to get oriented:
- 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:
The MC calculation of the residual lensing B modes for the template delensing method.
- 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.
The main result, constraints on from the methods compared. On this particular realization, Bayesian delensing achieves a 15% lower upper bound.
Same as above on four other realizations I ran.
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 .
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:
- He considered a higher noise level (ΔP = 2.3 vs. 0.68)
- He did not include 1/f noise
- 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.
Same as above, but for noise levels from Carron 2018 (and only 100deg², which I don’t expect matters)