## the three i’s of poverty

**T**oday I made a “quick” (10h door to door!) round trip visit to Marseille (by train) to take part in the PhD thesis defense (committee) of Edwin Fourrier-Nicolaï, which title was *Poverty, inequality and redistribution: an econometric approach*. While this was mainly a thesis in economics, meaning defending some theory on inequalities based on East German data, there were Bayesian components in the thesis that justified (to some extent!) my presence in the jury. Especially around mixture estimation by Gibbs sampling. (On which I started working almost exactly 30 years ago, when I joined Paris 6 and met Gilles Celeux and Jean Diebolt.) One intriguing [for me] question stemmed from this defense, namely the notion of a Bayesian estimation of a *three i’s of poverty* (TIP) curve. The three i’s stand for incidence, intensity, and inequality, as, introduced in Jenkins and Lambert (1997), this curve measure the average income loss from the poverty level for the *100p*% lower incomes, when p varies between 0 and 1. It thus depends on the distribution F of the incomes and when using a mixture distribution its computation requires a numerical cdf inversion to determine the income *p*-th quantile. A related question is thus on how to define a Bayesian estimate of the TIP curve. Using an average over the values of an MCMC sample does not sound absolutely satisfactory since the upper bound in the integral varies for each realisation of the parameter. The use of another estimate would however require a specific loss function, an issue not discussed in the thesis.

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