Last updated: 2024-11-11

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Rmd 657a21d Matthew Stephens 2024-11-11 workflowr::wflow_publish("flash_tree.Rmd")

Introduction

I wanted to outline the idea of applying flash to infer a tree-like structure.

Suppose we have a function flash_1pc(X,w) that takes as input an \(n \times p\) matrix \(X\) and an \(n\)-vector of weights \(w\) and produces a variational fit to the following model: \[X_{ij} = w_i(\mu_j + l_i f_j) + e_{ij}\] with \(l_i \sim g_l()\) and \(f_j \sim N(0,\sigma^2_f)\) (or maybe more general \(g_f()\)), and \(e_{ij} \sim N(0,\sigma^2)\).

By a fit we mean at least estimates \(\hat{\mu},\hat{l},\hat{f}\), and possibly estimates \(\hat{g}_l,\hat{\sigma}, \hat{\sigma_l}\). For convenience we also assume that it returns a matrix of expected residuals \(R_{ij} = X_{ij}-w_i(\hat{\mu}_j+\hat{l}_i \hat{f}_j)\).

Now consider applying this function recursively as follows:

flash_tree = function(X,w){
  if(w==0) return(list(fit_pc=NULL, fit_neg=NULL, fit_pos = NULL)) # end recursion
  
  fit_pc = flash_1pc(X,w)
  w = w * fit_pc$EL
  w_pos = ifelse(w>0, w, 0) # set nnegative weights to 0
  w_neg = ifelse(w<0, -w, 0) # set non-negative weights to 0, and make negative weights positive
  
  fit_pos = flash_tree(fit_pc$R,w_pos)
  fit_neg = flash_tree(fit_pc$R,w_neg)
  return(list(fit_pc = fit_pc, fit_neg = fit_neg, fit_pos = fit_pos))
}

\[Tree(X,w) = 1\mu + l_+ (f_+' + Tree(R, w=l_+)) + l_- (f_-' + Tree(R, w=l_-))\] The recursion ends at tips where the tree fit yields \(\hat{g}_l==\delta_0,\hat{l}==0\).