## Probabilities And Costs

In some situations, it is sufficient for a biosurveillance organization simply to know the posterior probability of an ongoing Cryptosporidium outbreak to make a decision, but in other situations (as in Glasgow), the costs and benefits of the available actions must be considered. If the posterior probability is near one, it is clear that action is required. If, the posterior probability is near zero, action is likely not warranted. A more formal decision analysis that includes costs and benefits of actions is needed to clarify what is the best action in the broad range of uncertainty between zero and one. If our goal is early detection, we will typically be operating in this gray area and require an understanding of the explicit tradeoffs between waiting and acting.

### 5. GENERALIZATION OF THE BAYESIAN WRAPPER METHOD

We illustrated the use of the Bayesian wrapper method to compute the posterior probability of a Cryptosporidium outbreak given a spike in sales of diarrhea remedies, P(C I A). In fact, we can find the expected characteristics of the possible outbreak by another application of Bayes' theorem. Specifically, we can compute the posterior distribution of S, D, and Y given A and C, and use it to find the expected size, duration, and start date of the outbreak. Even outside of a formal decision analysis, this information can provide insight into the source of the outbreak and guide management of the outbreak.

We can also use the Bayesian wrapper method when the differential diagnosis contains more than one disease. For example, we could consider that there are two possible causes for a spike in diarrhea remedies that is geographically mapped to a water distribution system: Cryptosporidium contamination and Giardia contamination.We could let C0 denote no outbreak, C1 denote Cryptosporidium contamination, and C2 denote Giardia contamination. We would specify prior probabilities for C0, C1, and C2. We would then compute P(AIC0), P(AIC1), and P(A IC2) as before, using nonoutbreak data and simulation. The set of outbreak characteristics and the priors for those characteristics could be different for the two diseases. Finally, we would use Bayes' theorem to calculate P(C0I A), P(C1I A), and P(C2I A).

The example illustrated how to use the Bayesian wrapper when the surveillance data consists of a single time series. Application of the method to multiple time series is more difficult because of the need to calculate P(A IC) and P(A IC). To calculate P(A IC), we need to model how an outbreak simultaneously affects multiple time series. Calculation of P(A IC) requires either modeling of multiple time series collected during nonoutbreak periods, or a substantial amount of surveillance data to enable empirical calculation of false-alarm rates.

6. SUMMARY_

The probabilistic interpretation of surveillance data is required for a formal analysis of a biosurveillance decision, and provides guidance for making decisions even outside of a formal decision analysis. Bayesian detection algorithms, such as BARD and PANDA, can provide this interpretation directly. However, most algorithms used in surveillance systems are non-Bayesian and do not provide posterior probabilities.

In this chapter we introduced the Bayesian wrapper method for computing posterior probabilities from the output of conventional (non-Bayesian) detection algorithms. We used the method to compute the posterior probability of a Cryptosporidium outbreak after seeing a spike in sales of diarrhea remedies. This probabilistic interpretation of surveillance data is one of the critical items that the Glasgow Incident Management Team was lacking, in addition to the costs and benefits of possible outcomes. Both of these items are needed to conduct a formal decision analysis. The following chapter discusses methods for estimating and modeling costs and benefits.