# Iterative Bayesian updating when some numerically estimated likelihoods are equal to zero

I am using Bayesian updating to derive the concentration of a contaminant from the results of a sequence of field sampling events. The laboratory methods used to analyze the water samples indicate presence/absence of the contaminant. So, the results of each field sampling iteration are expressed in terms of the fraction of water samples testing positive for the contaminant during that sampling event.

I have developed a model to simulate the probability of detecting the contaminant in a water sample at a given concentration in the environment. The model accounts for all of the steps of processing and analyzing water samples. I estimate the probability of detecting the contaminant by Monte Carlo simulation over a range of concentrations, 0 to 3000 copies/Liter. There are 601 discrete concentration intervals (0 copies/L and 600 intervals of 5 copies/L from 1 to 3000 copies/L).

The likelihoods for Bayesian updating, p(e|c), are obtained by post-processing these Monte Carlo simulation results to calculate the probability of observing e, some fraction of positive water samples given c, the concentration. The generation of likelihoods by Monte Carlo is a computationally intensive process (8 simulations of 50,000 samples, or 400,000 iterations).

I begin with a non-informative prior over a prospective range of concentration intervals. Initially, all prior probabilities are p(e) = 1/601. The posterior probability of the concentration interval given the evidence is calculated:

p(c|e) = p(e|c)p(c) / SUM_c p(e|c)p(e)

The posterior becomes the prior for the next iteration of updating - following the next sampling event. For some concentrations - particularly those that are low or high - the likelihoods of observing some evidence (fraction of positive samples) may be zero at some concentrations. As I update sequentially, the posterior, p(c|e), for these concentrations will also be zero. Therefore, as information accumulates over iterative sampling events, some concentrations become ruled out.

This is a problem where a high fraction of positive water samples is observed initially and this is followed by a series of sampling events with no detections. The posterior concentration estimates remain high because the lower concentration intervals were initially ruled out. This leads to an unreasonably (artificially) high concentration estimate.

My question is how to handle the zeros in this likelihood table. In most cases, the likelihoods obtained from Monte Carlo simulation are zero only because the sample size is small (400,000), not because the probability of observing the evidence is impossible. To correct this, I am tempted to substituted a low value (epsilon = 1e-7) for those likelihoods that are equal to zero. Is this legitimate, is there a precedence for this, and is there guidance for choosing epsilon?

• In a way, yes, there is a precedence for this, by analogy. The situation of sampling from a binomial distribution where all the obeservations are 0 (or 1) leads to an unrealistic ML estimate. A common "fix" to this was, at one time and possibly still, to add 1/2 success, 1/2 failure to the sample. In your case, this would lead to an epsilon of 1/800,000, (1/2 success and 400,000.5 failures to observe the event | c) or roughly 1e-6. – jbowman Jun 29 '15 at 17:10