Nasa’s new Earth Surface Mineral Dust Source Investigation instrument investigates how airborne dust particles affect the climate. Since July 2020, the instrument has revealed that more than a billion metric tonnes – the weight of 10,000 aircraft carriers – of mineral dust are carried through the atmosphere from deserts and dry regions to other areas on earth. One area that has been particularly badly affected is southeastern Europe, in particular Greece, Cyprus, and Turkey.
In a joint project, data scientists at the Cyprus Institute and Newcastle University are interested in the probability θ that particulate matter in the air, during and immediately following a dust storm, will result in an environmental health warning being issued for certain at-risk groups (i.e. people suffering from chronic lung conditions such as asthma).
(b) Of the last ten dust storms to affect southeastern Europe, four resulted in environmental health warnings (in particular, the storm affecting Crete and Cyprus in summer 2021).
(i) Assuming these storms occur independently, write down the likelihood function for θ.
(ii) Find the posterior distribution for θ, and use plots/sketches and numerical summaries to help describe changes in beliefs about θ in light of this information.
(c) Find the posterior distribution for θ assuming an appropriate ignorance prior. How does this compare to the posterior obtained using the informative prior?