# Uncertainty on zero counts for binned result

I do a counting experiment where I count observations as a function of two float parameters $x_1$ and $x_2$. This leads to a two-dimensional histogram where each bin corresponds to the number of observations with $x_1$ and $x_2$ in some range.

I now see a lot of bins with zero counts in them (even though more counts are in principle possible) and wonder how to best describe the uncertainty in these bins.

Knowing that I had $N$ observations which can fall into $k$ bins, I could assign an "expected rate" $n=\frac{N}{k}$ per bin and calculate the uncertainty with Poisson statistic. I however also know that my underlying process does not distribute events flat in $x_1$ and $x_2$ (but with a distribution I would like to extract from the measurement), how can I consistently assign an uncertainty in zero-count bins?

I do not want to rebin my data or drop these zero bins to not loose information.

• What do you mean by assigning an uncertainty? Do you want something like a standard error or confidence interval for each bin? Nov 20 '10 at 19:43
• @Aniko: Yes, I am looking for something like a confidence interval for my counts per bin which propagate nicely when I project out parts of my 2D histogram into 1D. Nov 24 '10 at 21:42

• That my null hypothesis is flat in $x_1$ and $x_2$ doesn't mean that it is flat in some transformed quantities which lead to my measurements being distributed flat in the available parameter space, right? Why do I not need to worry about this? And I am dealing with hundred millions of observations, so histograms are just easier to use for me. Nov 20 '10 at 3:07