Last updated: 2025-09-04
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| Rmd | 2e7d525 | Dave Tang | 2025-08-29 | Information theory |
According to Claude Shannon, information is present whenever a signal is transmitted from one place (sender) to another (receiver).
Information theory, founded by Shannon, studies the quantification, transmission, storage, and processing of information. At its core, it answers:
Key concepts include:
Mutual information between two random variables \(X\) and \(Y\) measures how much knowing one reduces uncertainty about the other.
\[ I(X;Y) = H(X) + H(Y) - H(X, Y) \]
or equivalently,
\[ I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X). \]
If \(I(X;Y) = 0\), \(X\) and \(Y\) are independent (no shared information); larger values mean stronger statistical dependence.
Install the dependencies required to render this document.
install.packages('mpmi')The dmi() function calculates MI and BCMI between a set
of discrete variables held as columns in a matrix. It also performs
jackknife bias correction and provides a z-score for the hypothesis of
no association. Also included are the *.pw functions that
calculate MI between two vectors only. The *njk functions
do not perform the jackknife and are therefore faster.
The results of dmi() are in many ways similar to a
correlation matrix, with each row and column index corresponding to a
given variable.
Exploring a group of categorical variables (from the examples in the
documentation of the dmi() function).
cyl - Number of cylindersvs - Engine (0 = V-shaped, 1 = straight)am - Transmission (0 = automatic, 1 = manual)gear - Number of forward gearscarb - Number of carburetors (a device used by a
gasoline internal combustion engine to control and mix air and fuel
entering the engine)my_vars <- c("cyl","vs","am","gear","carb")
dat <- mtcars[, my_vars]
discresults <- dmi(dat)
add_names <- function(res, names){
purrr::map(res, \(x){
row.names(x) <- names
colnames(x) <- names
x
})
}
add_names(discresults, my_vars)$mi
cyl vs am gear carb
cyl 1.0612040 0.43120940 0.14523133 0.3634430 0.5097002
vs 0.4312094 0.68531421 0.01417347 0.2036022 0.3123300
am 0.1452313 0.01417347 0.67546458 0.4367718 0.1248672
gear 0.3634430 0.20360224 0.43677177 1.0130227 0.2391776
carb 0.5097002 0.31232996 0.12486719 0.2391776 1.4979575
$bcmi
cyl vs am gear carb
cyl 1.0939730 0.397633050 0.105802510 0.2755075 0.48789448
vs 0.3976330 0.701457431 -0.003241008 0.1510687 0.29175135
am 0.1058025 -0.003241008 0.691622603 0.4355686 0.08710974
gear 0.2755075 0.151068658 0.435568574 1.0460800 0.16759348
carb 0.4878945 0.291751354 0.087109744 0.1675935 1.61116674
$zvalues
cyl vs am gear carb
cyl 21.798246 3.3933783 1.0582216 2.244308 7.474051
vs 3.393378 30.3263950 -0.1011464 1.223818 3.409049
am 1.058222 -0.1011464 19.9920905 5.522984 1.381430
gear 2.244308 1.2238177 5.5229835 14.478527 1.583226
carb 7.474051 3.4090490 1.3814296 1.583226 10.791836Two random variables.
set.seed(1984)
n <- 1000
X <- rbinom(n, 1, 0.5)
Y <- rbinom(n, 1, 0.5)
xy <- c('X', 'Y')
my_mat <- matrix(data = c(X,Y), nrow = n)
add_names(dmi(my_mat), xy)$mi
X Y
X 0.6926971130 0.0008345847
Y 0.0008345847 0.6923469671
$bcmi
X Y
X 0.6931976142 0.0003330715
Y 0.0003330715 0.6928474687
$zvalues
X Y
X 729.7075031 0.2572701
Y 0.2572701 547.067852580% of the time make Y the same as X; the other 20% of the time make Y 1 less than X.
set.seed(1984)
n <- 1000
X <- rbinom(n, 1, 0.5)
Y <- ifelse(runif(n) < 0.8, X, 1 - X)
xy <- c('X', 'Y')
my_mat <- matrix(data = c(X,Y), nrow = n)
add_names(dmi(my_mat), xy)$mi
X Y
X 0.6926971 0.2080063
Y 0.2080063 0.6910978
$bcmi
X Y
X 0.6931976 0.2075030
Y 0.2075030 0.6915983
$zvalues
X Y
X 729.7075 11.4776
Y 11.4776 341.4429
sessionInfo()R version 4.5.0 (2025-04-11)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.2 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
time zone: Etc/UTC
tzcode source: system (glibc)
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] mpmi_0.43.2.1 KernSmooth_2.23-26 lubridate_1.9.4 forcats_1.0.0
[5] stringr_1.5.1 dplyr_1.1.4 purrr_1.0.4 readr_2.1.5
[9] tidyr_1.3.1 tibble_3.2.1 ggplot2_3.5.2 tidyverse_2.0.0
[13] workflowr_1.7.1
loaded via a namespace (and not attached):
[1] sass_0.4.10 generics_0.1.4 stringi_1.8.7 hms_1.1.3
[5] digest_0.6.37 magrittr_2.0.3 timechange_0.3.0 evaluate_1.0.3
[9] grid_4.5.0 RColorBrewer_1.1-3 fastmap_1.2.0 rprojroot_2.0.4
[13] jsonlite_2.0.0 processx_3.8.6 whisker_0.4.1 ps_1.9.1
[17] promises_1.3.2 httr_1.4.7 scales_1.4.0 jquerylib_0.1.4
[21] cli_3.6.5 rlang_1.1.6 withr_3.0.2 cachem_1.1.0
[25] yaml_2.3.10 tools_4.5.0 tzdb_0.5.0 httpuv_1.6.16
[29] vctrs_0.6.5 R6_2.6.1 lifecycle_1.0.4 git2r_0.36.2
[33] fs_1.6.6 pkgconfig_2.0.3 callr_3.7.6 pillar_1.10.2
[37] bslib_0.9.0 later_1.4.2 gtable_0.3.6 glue_1.8.0
[41] Rcpp_1.0.14 xfun_0.52 tidyselect_1.2.1 rstudioapi_0.17.1
[45] knitr_1.50 farver_2.1.2 htmltools_0.5.8.1 rmarkdown_2.29
[49] compiler_4.5.0 getPass_0.2-4