# Bayesian vs Template Delensing

Marius Millea, April 26, 2020

## Contents

## 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:

## 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.

### 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.

#### Fig 4

Same as above on four other realizations I ran.

#### 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 .

## 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.

#### Fig 6

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