# Analytic approximation for r likelihood

*Colin Bischoff, 2018-11-16* // *updated 2018-11-20*

In a previous posting, I ran BICEP/Keck-style CosmoMC likelihood analysis for bandpowers corresponding to the model expectation value for r=0.003 + foregrounds + lensing residual. This allowed me to calculate the detection significance for r in those particular scenarios: experiment config 04b (Chile sky coverage) with residual A_{L}=0.27 or 0.337 and experiment config 04c (Pole sky coverage) with residual A_{L}=0.081 or 0.106. Victor's Fisher analysis allows us to calculate σ(r) for other scenarios, but we would like to estimate statistics like detection significance which rely on the non-Gaussian shape of the likelihood.

My ansatz is that the shape of the r likelihood can be well described with the H-L likelihood (Hamimeche & Lewis; PRD 77, 10, 103013; 2008). That likelihood is meant to describe CMB power spectra, but the BB spectrum is linear in r (more or less) so we might expect this choice to work well. For a one-dimensional likelihood with r as the only parameter, the form of the H-L likelihood simplifies (scalar multiplication commutes) and we can write it as

-log(L) = ( x - log x - 1 ) * ( r_{f}+ N )^{2}/ σ^{2}x = ( r_{ML}+ N ) / ( r + N )

where

`r`is the likelihood parameter,`r`is the maximum-likelihood r value,_{ML}`σ`is σ(r) calculated assuming a fiducial model with r =`r`,_{f}`N`is a "noise bias" that contains contributions from instrumental noise, residual foregrounds, and residual lensing.

In practice, if we want a representative likelihood curve for a particular value of r, we can use a Fisher code to calculate σ(r) then set `r _{ML}` =

`r`= r and

_{f}`σ`= σ(r). However, we still need to get an estimate of parameter

`N`from the CosmoMC-derived likelihoods.

For the four scenarios shown in my previous posting, I reran the CosmoMC likelihood with somewhat tighter convergence criteria and at a higher temperature to get a better measurement in the tails of the distribution. Then I fit each curve to the model by minimizing a K-S statistic. The results are shown in the following table, along with detection significance calculated both from the CosmoMC likelihood and from the analytic fit.

The table also includes a degrees of freedom statistic calculated as

k = 2 * ( r_{ML}+ N )^{2}/ σ^{2}

We might expect that this parameter should come out with a common value for the two Chile scenarios and a common value for the two Pole scenarios, with a ratio that corresponds to the relative sky area. I do find that both Pole scenarios correspond to `k` ~ 525 but I find very high and inconsistent values of `k` for the two Chile scenarios. I think the reason is that parameter `N` mostly affects the skewness of the distribution. The Pole likelihoods have significant skewness and I get a reliable fit. The Chile likelihoods have less skewness, so there is equal preference for any large value of `N`. Perhaps the best estimate of `k` for the Chile mask would be to start from the Pole mask value of `k`, then multiply by some map-derived estimate of the relative sky areas.

UPDATE 2018-11-20: I added new columns to the table and new lines to the figures showing how the likelihood shape and detection significance change if we fixkto 525 for Pole and 10000 for Chile.

Using the degrees of freedom parameter, we can write down an analytic model for the r likelihood by picking r, doing a Fisher calculation for σ(r), and then calculating `N` from the above equation.

Site | A_{L} |
r_{ML} |
σ |
N |
k |
significance (original) | significance (fit) | k* |
significance (fix k=k*)
| |
---|---|---|---|---|---|---|---|---|---|---|

Chile | 0.270 | 0.00256 | 0.00102 | 0.195 | 28446 | 2.5 σ | 2.5 σ | 10000 | 2.6 | |

Chile | 0.337 | 0.00255 | 0.00112 | 0.076 | 9803 | 2.3 σ | 2.3 σ | 10000 | 2.3 | |

Pole | 0.081 | 0.00268 | 0.00080 | 0.010 | 492 | 3.8 σ | 4.0 σ | 525 | 3.9 | |

Pole | 0.106 | 0.00269 | 0.00088 | 0.012 | 569 | 3.5 σ | 3.5 σ | 525 | 3.5 |

The figures below show the CosmoMC likelihood (blue) and the analytic model (orange) for the four scenarios that I used. The lower panels of each figure shows the fractional difference between the original likelihood and the model, which does increase out in the tails.