Biostatistics

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By J.L. Schafer

Offers a unified, Bayesian method of the research of incomplete multivariate facts, protecting datasets within which the variables are non-stop, specific or either. comprises genuine info examples and sensible suggestion.

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Extra resources for Analysis of Incomplete Multivariate Data

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In the above example, a particular factorization of the observed data likelihood enabled us to express the posterior in a tractable form. This will not always be the case. One cannot always factor the observed-data likelihood into complete-data likelihoods whose parameters are distinct. The techniques of Markov chain Monte Carlo to be introduced in Chapter 3 will free us from many of the constraints of mathematical tractability, allowing us to create random draws from the observed-data posterior whether or not it can be written in a tractable form.

When viewed as a probability distribution it summarizes knowledge about Ymis, for any assumed value of θ, and when viewed as a function of θ it conveys the evidence about θ contained in Ymis beyond that already provided by Yobs. 2) over the predictive distribution P(Ymis|Yobs,θ , where θ is a preliminary estimate of the unknown parameter. 3) where ( ) ∫ ( ) Q θ | θ (t ) = l(θ | Y )P Ymis | Yobs , θ (t ) dYmis and ( ) ∫ ( ) H θ | θ (t ) = log P(Ymis | Yobs , θ )P Ymis | Yobs , θ (t ) dYmis. A central result of Dempster, Laird, and Rubin (1977) is that if (t+1) (t) be the value of θ that maximizes Q(θ|θ ), then we let θ θ (t+1) is a better estimate than θ in the sense that its observed(t) data loglikelihood is at least as high as that of θ , (t) ( ) ( ) l θ (t +1) | Yobs ≥ l θ (t ) | Yobs .

The key ideas behind EM and data augmentation are the same: to solve a difficult incomplete-data problem by repeatedly solving tractable complete-data problems. As a result, the two methods share many features in common, and their implementation in specific examples is often remarkably similar. In this chapter, EM and data augmentation are introduced together to highlight the similarities between them. 1 Definition EM capitalizes on the interdependence between missing data Ymis and parameters θ .

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