Calculate Confidence Intervals for Poisson Processes with Backgrounds

2014 March 24

For some recent X-ray work, I’ve wanted to compute confidence intervals on the brightness of a source given a known background brightness. This is straightforward when the quantities in question are measured continuously, but for faint X-ray sources you’re in the Poisson regime, and things get a little trickier. If you’ve detected 3 counts in timespan τ, and you expect that 1.2 of them come from the background, what’s the 95% confidence interval on the number of source counts?

Of course, the formalism for this has been worked out for a while. Kraft, Burrows, and Nousek (1991) describe the fairly canonical (222 citations) approach. Their paper gives a lot of tables for representative values, but the formalism isn’t that complicated, so I thought I’d go ahead and implement it so that I can get values for arbitrary inputs.

Well, I wrote it, and I thought I’d share it in case anyone wants to do the same calculation. Here it is — in Python of course. There are a few subtleties but overall the calculation is indeed pretty straightforward. I’ve checked against the tables in KBN91 and everything seems hunky-dory. Usage is simple:

```
from pwkit.kbn_conf import kbn_conf
n = 3 # number of observed counts
b = 1.2 # expected number of background
counts cl = 0.95 # confidence limit
source_rate_lo, source_rate_hi = kbn_conf(n, b, cl)
# here, source_rate_lo = 0, source_rate_hi = 6.61 -- we have an upper limit on
# the source count rate
```

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