Added p_astro function

[wbenoit26]

Adds a function to calculate p_astro given a detection statistic, an astrophysical event rate, and the data products of an injection campaign.

To-do:

• [x] The function calculating nb can end up requiring a lot of memory. E.g., if we want the FAR of 1e6 foreground events and there are 4e6 background events, which is roughly typical, than it requires ~4 TB of memory. On the other hand, having this function vectorized makes it faster for large numbers of events. We'll need some kind of workaround if we want to calculate p_astro of all the events in an injection campaign.
• [x] Is this the best place for this function to live? Might be better over in utils
• [ ] What's the appropriate event rate to be using? Doesn't make a huge difference to the SV at p_astro > 0.5, but would be good to establish roughly how this number is chosen. I've been using 30 Gpc^-3 yr^-1 based on Table 4 of this paper

[EthanMarx]

Very sweet! Have you been able to try it out on some events / injections?

• for the nb calc, we should can probably just chunk it up into smaller pieces.

• I think this is a fine location for it now

[wbenoit26]

Yeah, I looked at a subset of an injection set. The curve looks pretty reasonable. Not totally monotonic because of the discreetness of the events, but I think that's fine, and there may be something more sophisticated we can do to avoid that. I'm going to calculate the SV for the re-weighted distribution and compare that to the catalog volumes to see whether we're roughly where we should be

[EthanMarx]

That looks great. How about many event's with p_astro > 0.5 in the 0lag would this correspond to?

Also, can you explain why theres some non-monotonicity?

[wbenoit26]

Looks like about 90, which is a little higher than I'd expect, but not crazy far off.

After a detection statistic of about 2.5, and especially at even higher values, there aren't very many background events relative to the number of foreground events. So as the detection statistic increases, the FAR remains constant for a short period while the sensitive volume decreases, which results in a net decrease to p_astro

[wbenoit26]

Likewise, our SV at p_astro > 0.5 is also a bit higher than expected, but not too far off. We're at 14.9, while PyCBC-BBH is at 10.5. I need to think some more about the details here, but I think to first order these results make sense.

[wbenoit26]

@EthanMarx I think this is good to be merged in. I'll integrate the calculation into the plotting code as I make updates there

[EthanMarx]

Great looks good. Not gonna make you do this here but I think we all should get back in the habit of adding tests for new functionality.

[EthanMarx]

Maybe also add in the event rate we are using as a constant somewhere?

[wbenoit26]

Good point on the tests, I'll add those in a separate PR. Not sure yet what the rate should be - I'll add that once I have a better understanding of how that value gets chosen