Contents

• Introduction
• Rough reasoning based on expectations
  • Setting up the problem
  • Fraction of sampled positives being really infected or not
  • Fraction of infectees in the positive sub-sample
  • Estimating the proportion of infectees in the population
  • Moving to probabilistic considerations
  • Summing up
  
  
• Probability of infected, in the light of the test result and of other relevant information
  • Bayes' rule at work
  • Initial odds, final odds and Bayes' factor
  • What do we learn by a second test?
  
  
• Uncertainty about and
  • From to : Bayes' rule applied to `numbers'
  • Conjugate priors
  • Expected value or most probable value of and ?
  • Effect of the uncertainties on and on the probabilities of interest
  • Adding also the uncertainty about
  • Uncertainty about InfPos and NoInfNeg?
  
  
• Predicting the number of positives resulting from testing a sample
  • Expected number of positives and its standard uncertainty
  • Taking into account the uncertainty on and
  • General considerations on the approximated evaluation of by Eq. ()
  
  
• Sampling a population
  • Proportion of infected individuals in the random sample - Binomial and hypergeometric distributions
  • Expected number of positives sampling of a population (assuming exact values of and )
  • Detailed study of the four contributions to
  • Balance between statistical and systematic contributions to the uncertainty on
  
  
• Measurability of
  • Probabilistic model
  • Monte Carlo estimates of and
  • Resolution power
  • Predicting the fractions of positives obtained sampling two different populations
  
  
• Inferring from the observed number of positives in the sample
  • From the general problem to its implementation in JAGS
  • Inferring and with our `standard parameters'
  • Dependence on our knowledge concerning and
  • Quality of the inference as a function of the sample size and of the fraction of positives in sample
  • Updated knowledge of and in the case of `anomalous' number of positives
  • Inferring the proportions of infectees in two different populations
  • Which priors?
  
  
• Exact evaluation of
  • Setting up the problem
  • Normalization factor and other moments of interest
  • Result and comparison with JAGS
  • More remarks on the role of priors
  
  
• Conclusions
• Bibliography

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Appendix A - Some remarks on `Bayes' formulae' . . . . . . . . . 99
Appendix B - R and JAGS code . . . . . . . . . . . . . . . . . 103