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Combining expert probabilities using the product of odds

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Abstract

We resolve a useful formulation of the question how a statistician can coherently incorporate the information in a consulted expert’s probability assessment for an event into a personal posterior probability assertion. Using a framework that recognises the total information available as composed of units available only to each of them along with units available to both, we show: that a sufficient statistic for all the information available to both the expert and the statistician is the product of their odds ratios in favour of the event; that the geometric mean of their two probabilities specifies a contour of pairs of assertions in the unit-square that yield the same posterior probability; that the information-combining function is parameterised by an unknown probability for the event conditioned only on the unspecified information common to both the statistician and the expert; and that an assessable mixing distribution over this unspecified probability allows an integrable mixture distribution to represent a computable posterior probability. The exact results allow the identification of the subclass of coherent probabilities that are externally Bayesian operators. This subclass is equivalent to the class of combining functions that honour the principles of uniformity and compromise.

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References

  • Clemen R. T. (1987) Combining overlapping information. Management Science 33(3): 373–380

    Article  Google Scholar 

  • Clemen R. T., Winkler R. L. (1990) Unanimity and compromise among probability forecasters. Management Science 36(7): 767–779

    Article  Google Scholar 

  • Clemen R.T., Winkler R.L. (1999) Combining probability distributions from experts in risk analysis. Risk Analysis 19: 187–203

    Google Scholar 

  • Dawid A., DeGroot M., Morterra J. (1995) Coherent combination of expert opinion (with discussion). Test 5: 263–313

    Article  Google Scholar 

  • Di Bacco M., Lad F. (2002) Assessing the value of a second opinion: The role and structure of exchangeability. Annals of Mathematics and Artificial Intelligence 35(1–4): 227–252

    Google Scholar 

  • Di Bacco, M., Frederic, P., & Lad, F. (2003). Learning from the probability assertions of experts, Tech. rep., Mathematics and Statistics department at Canterbury University (NZ). http://www.math.canterbury.ac.nz/research/ucdms2003n6.pdf.

  • Dickey J. (1980) Beliefs about beliefs a theory of stochastic assessments of subjective probabilities. In: Bernardo J., DeGroot M. H., Lindley D. V., Smith A. F. M. (eds) Bayesian statistics. University Press, Valencia, pp 471–487

    Google Scholar 

  • Frederic P., Catellani M. (2003) Statistical analysis of an industrial distillation process. Statistica Applicata 15(2): 263–273

    Google Scholar 

  • Frederic P., Lad F. (2008) Two moments of the logitnormal distribution. Communication in Statistics: Simulation and Computation 37: 1263–1269

    Article  Google Scholar 

  • French S., Rios Insua D. (2000) Statistical decision theory. Arnold, London

    Google Scholar 

  • Genest C., McConway K., Schervish M. (1986) Characterization of externally Bayesian pooling operators. Annals of Statistics 145: 487–501

    Article  Google Scholar 

  • Good I. (1979) On the combination of judgements concerning quantiles of a distribution with potential application to the estimation of mineral resources. Journal of Statistical Computation and Simulation 9: 77–79

    Article  Google Scholar 

  • Lindley, D. (1985). Reconciliation of discrete probability distributions, with discussion. In J. M. Bernardo et al. (Eds.), Bayesian Statistics 2: Proceedings of the Second Valencia International Meeting, 1983, Amsterdam, North-Holland (pp. 375–390).

  • Lindley D., Tversky A., Brown R. (1979) On the reconciliation of probability. Journal of the Royal Statistical Society A 142: 146–180

    Article  Google Scholar 

  • Madansky, A. (1964). Externally Bayesian groups. Tech. rep., Rand Corporation Memo Rm-4141-PR, Santa Monica: Rand.

  • McConway K. (1981) Marginalisation and linear opinion pools. Journal of the American Statistical Association 76: 410–414

    Article  Google Scholar 

  • Morris A. (1974) Decision analysis expert use. Management Science 20: 1233–1244

    Article  Google Scholar 

  • Morris A. (1986) Observation on expert aggregation. Management Science 32: 321–328

    Article  Google Scholar 

  • O’Hagan A. (2006) Research in elicitation. In: Upadhyay S. K., Singh U., Dey D. K. (eds) Bayesian statistics and its applications. Anamaya, New Delhi, pp 375–382

    Google Scholar 

  • Winkler R. (1968) Consensus of subjective probability distributions. Management Science 15: 61–75

    Article  Google Scholar 

  • Winkler R. (1981) Combining probability distributions from dependent information sources. Management Science 27: 479–488

    Article  Google Scholar 

  • Zeckhauser R. (1971) Combining overlapping information. Journal of the American Statistical Association 66: 91–92

    Article  Google Scholar 

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Correspondence to Patrizio Frederic.

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Frederic, P., Di Bacco, M. & Lad, F. Combining expert probabilities using the product of odds. Theory Decis 73, 605–619 (2012). https://doi.org/10.1007/s11238-012-9320-9

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