# Picking a probability distribution for observed intensities

I have an experiment that measures "intensity" (in this case, electron density of a molecule) on a grid. The values it gives are non-negative,. I'd like to write a likelihood for this observation given a model calculated on the same grid, assuming the grid measurements are independent.

Since the values are non-negative, and I would like to be somewhat tolerant of large errors, I originally thought I'd choose a lognormal distribution (here for a single data point):

$$L(I_{exp}|I_{mod})=\frac{1}{\sqrt{2\pi}\sigma I_{exp}}\exp{\left[-\frac{\log^2(I_{exp}/I_{mod})}{2\sigma^2}\right]}$$

Here's the problem: the data values $I_{exp}$ are sometimes zero, because the experimental device cannot go below some degree of precision. But the lognormal goes to $0$ as $I_{exp}\rightarrow0$ So at the low values it might be more accurate to pick a truncated normal distribution.

Are there any distributions that 1) have non-zero probability at 0 and 2) are more tolerant of large errors than a normal pdf?

1) You could try a left-censored model since you know that the value is below the threshhold but you don't know what value it has, by dealing with the censored-data likelihood.

2) You might try a zero-inflated (ZI) model, such as a ZI-lognormal or ZI-gamma. These rely on using (say) a logistic model for the 0/non-0 and some continuous model thereafter. Better still would be to deal with a ZI-truncated lognormal (since that could account for the lack of observations between 0 and the threshold), but it might not make much difference.

• Yes, thinking of it as a data-censoring problem makes sense. The only issue in my case is that there's no way of determining exactly what the censor threshold is. Do you know any examples where the exact censor value was sampled? Dec 18 '14 at 1:04
• By "where the exact censor value was sampled" do you mean "was estimated from the sample"? Dec 18 '14 at 1:30
• Yes, exactly. Naively I would just guess the censor value to be the lowest non-zero measured point, but this is probably an over-estimate. Would be interesting if it were possible to estimate the censor threshold in a Bayesian manner. Dec 18 '14 at 1:32