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SEQUENTIAL IMPUTATIONS AND BAYESIAN MISSING DATA PROBLEMS AUGUSTING KONG, JUN LIU WING HUNG WONG

SEQUENTIAL IMPUTATIONS AND BAYESIAN MISSING DATA PROBLEMS AUGUSTING KONG, JUN LIU WING HUNG WONG JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION MARCH 1994 VOL 89 NO. 425. Setting. X=(x1,…xn)=(y1, z1,…yn, zn)=(Y, Z) If an observation l is complete, then yl=xl. Goal. Importance sampling.

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SEQUENTIAL IMPUTATIONS AND BAYESIAN MISSING DATA PROBLEMS AUGUSTING KONG, JUN LIU WING HUNG WONG

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  1. SEQUENTIAL IMPUTATIONS AND BAYESIAN MISSING DATA PROBLEMS • AUGUSTING KONG, JUN LIU WING HUNG WONG • JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION MARCH 1994 VOL 89 NO. 425

  2. Setting • X=(x1,…xn)=(y1, z1,…yn, zn)=(Y, Z) • If an observation l is complete, then yl=xl

  3. Goal

  4. Importance sampling • Draw m independent copies of Z’s from the conditional distribution p(Z|Y) and then approximate

  5. problem • Drawing from p(Z|Y) directly is usually difficult • Gibbs sampler or data augmentation do this approximately by iterations

  6. Sequential Imputation • Step 1: • Draw zt* from the conditional distribution p(zt|y1, z1*,…yt-1, zt-1*, yt). Notice that the zt*’s had to be drawn sequentially, because each zt* is drawn conditioned on the previously imputed missing part z1*,…,zt-1*

  7. Sequential Imputation • Step2: • Compute the predictive probabilities p(yt|y1, z1*,…,yt-1, zt-1*) and • wt=wt-1 p(yt|y1, z1*,…,yt-1, zt-1*) • Let w=wn, so that • W=p(y1)π p(yt|y1, z1*,…,yt-1, zt-1*) , for t=2…n

  8. Sequential Imputation • Step1 and step2 are done repeatedly and independently for m times • Let the results be denoted by Z*(1), Z*(2),…Z*(m) and w(1),…w(2), where Z*(j)=(z1*(j),…zn*(j)) for j=1…m

  9. Sequential Imputation • Posterior distribution:

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