Generative models
2024-08-07
Last updated: 2024-08-07
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Knit directory: muse/
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Discrete example
If we know the rules (parameters) of a mechanism, then even if the outcomes are random, we can generate probabilities of any event by using computations and standard probability laws.
Consider that mutations along the genome of Human Immunodeficiency Virus (HIV) occur at random with a rate of \(5 \times 10^{-4}\) per nucleotide per replication cycle. This means that after one cycle, the number of mutations in a genome of about \(10^4\) = 10,000 nucleotides will follow a Poisson distribution with rate 5.
This probability model predicts that the number of mutations over one replication cycle will be close to 5 and that the variability of this estimate is \(\sqrt{5}\) (the standard error). We now have baseline reference values for both the number of mutations we expect to see in a typical HIV strain and its variability.
If we want to know how often 3 mutations could occur under the
Poisson(5) model, we can use the dpois() function to
generate the probability of seeing x = 3 events, taking the value of the
rate parameter of the Poisson distribution, called lambda
(\(\lambda\)), to be 5.
dpois(x = 3, lambda = 5)[1] 0.1403739The output above says that the chance of seeing exactly three events is around 0.14 or about 1 in 7.
Probabilities of a range of values.
dpois(x = 0:12, lambda = 5) |>
barplot(names.arg = 0:12, col = 2)
Mathematical theory tells us that the Poisson probability of seeing \(x\) is given by:
\[ p = \frac{e^{-\lambda} \lambda^x}{x!} \]
my_dpois <- function(x, lambda){
e <- exp(1)
((e^-lambda)*(lambda^x))/factorial(x)
}
my_dpois(3, 5)[1] 0.1403739dpois(3, 5)[1] 0.1403739The Poisson distribution is a good model for rare events such as mutations. Other useful probability models for discrete events are the Bernoulli, binomial, and multinomial distributions.
sessionInfo()R version 4.4.0 (2024-04-24)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 22.04.4 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-r0.3.20.so; LAPACK version 3.10.0
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time zone: Etc/UTC
tzcode source: system (glibc)
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1 dplyr_1.1.4
[5] purrr_1.0.2 readr_2.1.5 tidyr_1.3.1 tibble_3.2.1
[9] ggplot2_3.5.1 tidyverse_2.0.0 workflowr_1.7.1
loaded via a namespace (and not attached):
[1] sass_0.4.9 utf8_1.2.4 generics_0.1.3 stringi_1.8.4
[5] hms_1.1.3 digest_0.6.35 magrittr_2.0.3 timechange_0.3.0
[9] evaluate_0.24.0 grid_4.4.0 fastmap_1.2.0 rprojroot_2.0.4
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[17] promises_1.3.0 httr_1.4.7 fansi_1.0.6 scales_1.3.0
[21] jquerylib_0.1.4 cli_3.6.2 rlang_1.1.4 munsell_0.5.1
[25] withr_3.0.0 cachem_1.1.0 yaml_2.3.8 tools_4.4.0
[29] tzdb_0.4.0 colorspace_2.1-0 httpuv_1.6.15 vctrs_0.6.5
[33] R6_2.5.1 lifecycle_1.0.4 git2r_0.33.0 fs_1.6.4
[37] pkgconfig_2.0.3 callr_3.7.6 pillar_1.9.0 bslib_0.7.0
[41] later_1.3.2 gtable_0.3.5 glue_1.7.0 Rcpp_1.0.12
[45] highr_0.11 xfun_0.44 tidyselect_1.2.1 rstudioapi_0.16.0
[49] knitr_1.47 htmltools_0.5.8.1 rmarkdown_2.27 compiler_4.4.0
[53] getPass_0.2-4