2008-11-20

parameter expansion yay!

I was aware that the subject covariates in my hierarchical 1-dimensional ideal point model induced greater dependence in the chain. The reason is obvious – the full conditional update of the subject parameters θ now incorporates two pieces of information from the previous iteration (both η and β). This means much slower mixing chains.

Parameter expansion (px-da) actually improves mixing over the standard data augmentation scheme. Liu and Wu (1999) prove that convergence is always faster, but it still seems like magic. The method is nearly identical to the Marginal Data Augmentation of Meng & van Dyk that Simon Jackman just added to ideal, except that I sample the expansion parameter α on the residual variance of the latent data from an inverse-Gamma distribution whose prior parameters you can change. MDA fixes the prior value of the expansion parameter at 1 (equivalent to IG(a,b) as a→0).

I'm not sure when ADM and KQ plan the next MCMCpack release, but the default in MCMCirtHier1d is px=TRUE. The figure below shows the improvement in autocorrelation: for two chains of 10k iterations thinned every 20 samples, after a burnin of 50k iterations, the autocorrelation (which drops of really, really slowly for some subjects) drops below that even of the naïve model without PX (not shown). The black line is with PX, the red line without.

Parameter Expansion versus regular Data Augmentation

this week

From CSA 2k8
  • nappa
  • red bok choi
  • beets (big red ones, and tiny white ones with great greens)
  • baby black kale
  • dumpling squashes
  • radishes
  • apples
  • onion
  • hot peppers

last week

From CSA 2k8
  • chickens
  • turnips
  • parsley root with some parsley
  • carrots
  • collards
  • green cabbage
  • mesclun
  • apples
  • brussels sprouties!

2008-11-08

what to do with all this food…

  • [660] black chick peas, kala chana masala
  • [660] saag of collard, radish, beet greens
  • stir fried bok choi and celery with ‘brown sauce’
  • beet soup in acorn squashes
  • perhaps an apple pie?
  • saag paneer
  • [660] rajma with turnips
  • turnip and sweet potato gratin
  • salad with cold noodles, ginger-sesame sauce

fall box 2

From CSA 2k8
  • pumpkin
  • baby white beets (mostly for their delicious greens)
  • heirloom celery
  • radishes
  • red bok choi
  • more fucking collard greens
  • apples
  • red lettuce
  • onion
  • (three sisters) green garlic
  • (smits) mint
  • thyme
  • eggies
  • apples
  • sweet potatoes

last week (fall csa 1)

From CSA 2k8
  • carrots
  • collard greens
  • cabbage
  • apples
  • mesclun
  • brussels sprouts
  • eggs
  • daikons
  • parsley root
  • turnips
  • onion
  • acorn squash