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Introduction

Look at log of density function

\[g(x) = \sum_k w_k N(x;0,\sigma^2_k).\]

#'second derivative of log g(x) where g() is a mixture of Gaussian
#'@param x observation, a scalar.
#'@param w mixture weights.
#'@param s2 variances of mixture components.
lg_d2 = function(x,w,s2){
  n1 = sum(w/sqrt(2*pi*s2)*exp(-x^2/2/s2)*((x/s2)^2-1/s2))
  d = sum(w/sqrt(2*pi*s2)*exp(-x^2/2/s2))
  n2 = -sum(w/sqrt(2*pi*s2)*exp(-x^2/2/s2)*x/s2)
  return(n1/d - (n2/d)^2)
}

#'second derivative of log g(x) where g() is a mixture of Gaussian
#'x is a vector
lg_d2_vec = function(x,w,s2){
  n = length(x)
  out = rep(0,n)
  for(i in 1:n){
    out[i] = lg_d2(x[i],w,s2)
  }
  return(out)
}

#'density of log g(x)
#'x is a scalar
lg = function(x,w,s2){
  return(log(sum(w/sqrt(2*pi*s2)*exp(-x^2/2/s2))))
}

#'density of log g(x)
#'x is a vector
lg_vec = function(x,w,s2){
  n = length(x)
  out = rep(0,n)
  for(i in 1:n){
    out[i] = lg(x[i],w,s2)
  }
  return(out)
}

plot_lg = function(x,w,s2){
  plot(x,lg_vec(x,w,s2),type='l',xlab='x',ylab='log(g)',
       main=paste('w=(',paste(w,collapse = ", "),'); s2=(',paste(s2,collapse = ", "),')',sep=''))
}

plot_lg_d2 = function(x,w,s2){
  plot(x,lg_d2_vec(x,w,s2),type='l',xlab='x',ylab="(log g)''",
       main=paste('w=(',paste(w,collapse = ", "),'); s2=(',paste(s2,collapse = ", "),')',sep=''))
}

log concave and variance ratio

Fix \(w=(0.5,0.5)\), and smaller variance to be 1. Then vary the larger one from 1.1 to 4. Check \(max(\log''(g))\). If it’s smaller than 0, the \(\log g\) is concave.

Then we divide the both variances by 10, so the smaller one is 0.1, and plot the same figure.

Finally, we multiply both variances by 10, so the smaller variance is 10, and plot the same figure.

x = seq(-10,10,length.out = 1000)
w = c(0.5,0.5)
s2_list=seq(1.1,4,by=0.1)
out = c()
for(ss in s2_list){
  out = c(out,max(lg_d2_vec(x,w,c(1,ss))))
}

par(mfrow=c(1,3))

plot(s2_list,out,type='l',xlab='ratio',ylab="max(log''(g))")
abline(h=0,lty=2)


s2_list=seq(1.1,4,by=0.1)/10
out = c()
for(ss in s2_list){
  out = c(out,max(lg_d2_vec(x,w,c(0.1,ss))))
}
plot(s2_list,out,type='l',xlab='ratio',ylab="max(log''(g))")
abline(h=0,lty=2)

s2_list=seq(1.1,4,by=0.1)*10
out = c()
for(ss in s2_list){
  out = c(out,max(lg_d2_vec(x,w,c(10,ss))))
}
plot(s2_list,out,type='l',xlab='ratio',ylab="max(log''(g))")
abline(h=0,lty=2)

All lines have the same shape and the intersection points are at the same position. This confirms that with weights fixed, the concavity is related to the variance ratios.


sessionInfo()
R version 4.2.1 (2022-06-23 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 22000)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.utf8 
[2] LC_CTYPE=English_United States.utf8   
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.utf8    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.9       highr_0.9        compiler_4.2.1   pillar_1.8.1    
 [5] bslib_0.4.0      later_1.3.0      git2r_0.30.1     jquerylib_0.1.4 
 [9] tools_4.2.1      getPass_0.2-2    digest_0.6.29    jsonlite_1.8.0  
[13] evaluate_0.16    tibble_3.1.8     lifecycle_1.0.2  pkgconfig_2.0.3 
[17] rlang_1.0.5      cli_3.3.0        rstudioapi_0.14  yaml_2.3.5      
[21] xfun_0.32        fastmap_1.1.0    httr_1.4.4       stringr_1.4.1   
[25] knitr_1.40       fs_1.5.2         vctrs_0.4.1      sass_0.4.2      
[29] rprojroot_2.0.3  glue_1.6.2       R6_2.5.1         processx_3.7.0  
[33] fansi_1.0.3      rmarkdown_2.16   callr_3.7.2      magrittr_2.0.3  
[37] whisker_0.4      ps_1.7.1         promises_1.2.0.1 htmltools_0.5.3 
[41] httpuv_1.6.5     utf8_1.2.2       stringi_1.7.8    cachem_1.0.6