Epidemiology and Infection
نویسندگان
چکیده
Bayesian inference is based on the factorisation of the posterior joint probability function, according to Bayes’ theorem, into a prior distribution for the parameters θ and the likelihood of the data Y given the parameters. The aim of evidence synthesis is to estimate k “basic” parameters θ = {θ1, . . . , θk} from n datasets Y = {Y1, . . . , Yn}. The distribution of each dataset Yi is determined by parameters that may be expressed as functions of the basic parameters: Yi ∼ f(Yi | Gi(θ)) = f(Yi | θ). The function Gi is the identity function for a single component θj of θ, Gi(θ) = θj if the i th dataset directly informs θj. If the dataset indirectly informs multiple parameters, Gi is a more complex function of θ. If each data source is assumed independent, a likelihood function combining all data items may be constructed as the product of contributions from each data source, L(Y | θ) = ∏n i=1 L(Yi | θ). Additionally, other pre-existing knowledge may be added in the form of the prior distribution on θ, determined by additional hyper-parameters λ: f(θ | λ). Using the Bayesian approach, the joint posterior distribution of θ conditional on the available data and hyper-parameters is:
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