Message ID: 2571
Entry time: Wed May 13 18:07:32 2020
Reply to this: 2572

Author:

anchal

Type:

DailyProgress

Category:

NoiseBudget

Subject:

Bayesian Analysis

I did this analysis last with bare-bones method in CTN:2439. Now I've improved this much more. Following are some salient features:

Assuming Uniform prior distribution of Bulk Loss Angle since the overlap with Penn et al. is so low that our measurement is inconsistent with theirs ((5.33 +- 0.03) x 10^{-4} )if we take into account their extremely low standard deviation associated to bulk loss angle.

Assuming Normal Distributed prior distribution for Shear Loss Angle matching Penn et al. reported value of (2.6 +- 2.6) x 10^{-7}. This is done because we can faithfully infere only one of the two loss angles.

The likelihood function is estimated in the following manner:

Data cleaning:

Frequency points are identified between 50 Hz to 700 Hz where the derivative of Beat Note Frequency noise PSD with respect to frequency is less than 2.5 x 10^{-5} Hz^{2}/Hz^{2}..

This was just found empirically. This retains all low points in the data away from the noise peaks.

Measured noise Gaussian:

At each "clean" frequency point, a gaussian distribution of measured beat note frequency noise ASD is assumed.

This gaussian is assumed to have a mean of the corresponding measured 'median' value.

The standard deviation is equal to half of the difference between 15.865 percentile and 84.135 percentile points. These points correspond to mean +- standard deviation for a normal distribution

Estimated Gaussian and overlap:

For an iterable value of Bulk and Shear Loss Angle, total noise is estimated with estimated uncertainty. This gives a gaussian for the estimated noise.

The overlap of two Gaussians is calculated as the overlap area. This area which is 0 for no overlap and 1 for complete overlap is taken as the likelihood function.

However, any estimate of noise that goes above the measured nosie is given a likelihood of zero. Hence the likelihood function in the end looks like half gaussian.

The likelihood for different clean data points is multiplied together to get the final likelihood value.

The Product of prior distribution and likelihood function is taken as the Bayesian Inferred Probability (unnormalized).

The maximum of this distribution is taken as the most likely inferred values of the loss angles.

The standard deviation for the loss angles is calculated from the half-maximum points of this distribution.

Final results are calculated for data taken at 3 am on March 11th, 2020 as it was found to be the least noise measurement so far:

Bulk Loss Angle: (8.8 +- 0.5) x 10^{-4}.

Shear Loss Angle: (2.6 +- 2.85) x 10 ^{-7}.

Figures of the analysis are attached. I would like to know if I am doing something wrong in this analysis or if people have any suggestions to improve it.

The measurement instance used was taken with HEP filter on but at low. I expect to measure even lower noise with the filters completely off and optimizing the ISS as soon as I can go back to lab.

Other methods tried:

Mentioning these for the sake of completeness.

Tried using a prior distribution for Bulk Loss Angle as a gaussian from Penn et al. measured value. The likelihood function just became zero everywhere. So our measurements are not consistent at all. This is also because the error bars in their reported Bulk Loss Angle are extremely

Technically, the correct method for likelihood estimation would be following:

Using the mean () and standard deviation () of estimated total noise, the mean of the measured noise would be a gaussian distribution with mean and variance where N is the number of averaging in PSD calculation (600 in our case).

If standard deviation of the measured noise is , then would be a distribution with N-1 degrees of freedom.

These functions can be used to get the probability of observed mean and standard deviation in the measured noise with a prior distribution of the total estimated noise distribution.

I tried using this method for likelihood estimation and while it works for a single frequency point, it gives zero likelihood for multiple frequency points.

This indicated that the shape of the measured noise doesn't match well enough with the estimated noise to use this method. Hence, I went to the overlap method instead.