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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.

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    $\begingroup$ What do you mean by assigning an uncertainty? Do you want something like a standard error or confidence interval for each bin? $\endgroup$
    – Aniko
    Nov 20 '10 at 19:43
  • $\begingroup$ @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. $\endgroup$ Nov 24 '10 at 21:42
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Nothing wrong with what you've done so far. The null model includes only a constant, i.e. a flat event rate. Fitting more complex Poisson regression models will allow the expected value to vary. It's hard to tell what forms the more complex models should take as you've told us nothing about the source of the data, how much data you have, or the question(s) you wish to answer. Binning the data may help suggest what models are appropriate, but you may be better keeping continuous covariates continuous and fitting some smoother form (splines, fractional polynomials, local regression...).

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  • $\begingroup$ 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. $\endgroup$ Nov 20 '10 at 3:07
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Try running a nonparametric smoother (also called a kernel density estimate) over your data to estimate the expected value (and therefore proportion

If you do have covariate data, see how the nonparametric smooth compares to the regression model that onestop recommends. A parametric model is usually a lot less wiggly than a simple smoothed estimate.

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