Theme Song

1 テスト 1 (Assessment 1)

2 Introduction

合計点数 5点

2.1 1. Definition of Mixture Models

合計点数 5

2.2 1. 質問 1

Which one of the following is not the density of a well defined mixture distribution with support on the positive integers (1点)

\({\color{Green}f(x) = 0.5 \times \frac{e^{-1}}{x!} + 0.5 \times \frac{e^{-1}}{x!}}\)

正解 : This is not a well defined mixture since the first component does not correspond to a well defined probability mass function.

\(f(x) = 0.5 \times \frac{2^x e^{-2}}{x!} + 0.5 \times \frac{3^x e^{-3}}{x!}\)
\(f(x) = 0.45 \times 2^x \frac{ e^{-1}}{x!} + 0.55 \times \frac{3^x e^{-3}}{x!}\)

2.3 2. 質問 2

Consider a zero-inflated mixture that involves a point mass at 0 with weight 0.2, a Gamma distribution with mean 1, variance 2 and weight 0.5, and another Gamma distribution with mean 2, variance 4 and weight 0.3. What is the mean of this mixture? (1点)

正解 : \(E(X)=0.2×0+0.5×1+0.3×2=1.1\)

2.4 3. 質問 3

こちらに回答を入力 *1.060159, 0.07962572

不正解 : Recall that \[Var(X) = E(X^2) - [E(X)]^2\] and that expectations are linear functionals. In the videos, we showed that \[E(X^2) = \sum_{k=1}^K w_k \left[Var_{g_k}(X) + [E_{g_k}(X)]^2 \right]\] where \(g_k\) is the \(k\)th component of the mixture

2.5 4. 質問 4

True or False: A mixture of Gaussians of the form

\[f(x) = 0.3 \frac{1}{\sqrt{2\pi}} \exp\left\{ - \frac{x^2}{2} \right\} + 0.7 \frac{1}{\sqrt{2\pi}} \exp\left\{ - \frac{(x-4)^2}{2} \right\}\]

has a bimodal density. (1点)

True

正解 : A hint is that the means of the two components are quite well separated, which often leads to multimodal densities. The easiest way to verify this intuition is by plotting the density:

x = seq(-3, 7, length=100)
y = 0.3*dnorm(x, 0, 1) + 0.7*dnorm(x, 4, 1)
par(mar=c(4,4,1,1)+0.1)
plot(x, y, type="l", ylab="Density", las=1, lwd=2)

False

2.6 5. 質問 5

True or False: Consider a location mixture of normals

\[f(x) = \sum_{k=1}^{K} \omega_k \frac{1}{\sqrt{2\pi} \sigma} \exp \left\{ - \frac{ \left(x - \mu_k\right)^2}{2\sigma^2} \right\}\]

The following 3 constraints make all parameters fully identifiable:

The means \(\mu_1, \ldots, \mu_K\) should all be different.
No weight \(\omega_k\) is allowed to be zero.
The component are ordered based on the values of their means, i.e., the component with the smallest \(\mu_k\) is labeled component 1, the one with the second smallest \(\mu_k\) is labeled component 2, etc. (1点)

True

正解 : The three constraints are enough to ensure identifiability. The last one addresses label switching, while the first two address identifiability of the number of components in the mixture.

False

3 Appendix

3.1 Blooper

3.2 Documenting File Creation

It’s useful to record some information about how your file was created.

File creation date: 2021-05-09
File latest updated date: 2021-05-11
R version 4.0.5 (2021-03-31)
rmarkdown package version: 2.8
File version: 1.0.0
Author Profile: ®γσ, Eng Lian Hu
GitHub: Source Code
Additional session information:

suppressMessages(require('dplyr', quietly = TRUE))
suppressMessages(require('magrittr', quietly = TRUE))
suppressMessages(require('formattable', quietly = TRUE))
suppressMessages(require('knitr', quietly = TRUE))
suppressMessages(require('kableExtra', quietly = TRUE))

sys1 <- devtools::session_info()$platform %>% 
  unlist %>% data.frame(Category = names(.), session_info = .)
rownames(sys1) <- NULL

sys2 <- data.frame(Sys.info()) %>% 
  dplyr::mutate(Category = rownames(.)) %>% .[2:1]
names(sys2)[2] <- c('Sys.info')
rownames(sys2) <- NULL

if (nrow(sys1) == 9 & nrow(sys2) == 8) {
  sys2 %<>% rbind(., data.frame(
  Category = 'Current time', 
  Sys.info = paste(as.character(lubridate::now('Asia/Tokyo')), 'JST🗾')))
} else {
  sys1 %<>% rbind(., data.frame(
  Category = 'Current time', 
  session_info = paste(as.character(lubridate::now('Asia/Tokyo')), 'JST🗾')))
}

sys <- cbind(sys1, sys2) %>% 
  kbl(caption = 'Additional session information:') %>% 
  kable_styling(bootstrap_options = c('striped', 'hover', 'condensed', 'responsive')) %>% 
  row_spec(0, background = 'DimGrey', color = 'yellow') %>% 
  column_spec(1, background = 'CornflowerBlue', color = 'red') %>% 
  column_spec(2, background = 'grey', color = 'black') %>% 
  column_spec(3, background = 'CornflowerBlue', color = 'blue') %>% 
  column_spec(4, background = 'grey', color = 'white') %>% 
  row_spec(9, bold = T, color = 'yellow', background = '#D7261E')

rm(sys1, sys2)
sys

Additional session information:
Category	session_info	Category	Sys.info
version	R version 4.0.5 (2021-03-31)	sysname	Linux
os	Ubuntu 20.04.2 LTS	release	5.8.0-52-generic
system	x86_64, linux-gnu	version	#59~20.04.1-Ubuntu SMP Fri Apr 30 16:10:51 UTC 2021
ui	X11	nodename	Scibrokes-Trading
language	en	machine	x86_64
collate	C	login	englianhu
ctype	en_US.UTF-8	user	englianhu
tz	Asia/Tokyo	effective_user	englianhu
date	2021-05-11	Current time	2021-05-11 01:30:18 JST🗾

Bayesian Statistics: Mixture Models (Assessment 1)

®γσ, Lian Hu ®

2021-05-11