Last updated: 2017-03-06
Code version: c7339fc This illustrates how the prior, likelihood, and posterior behave for inference for a normal mean (\(\mu\)) from normal-distributed data, with a conjugate prior on \(\mu\).
Specifically the prior on \(\mu\) is N(\(\mu_0\), \(\tau_0^2\)) [dotted line] and the data is sampled from a normal distribution N(\(\mu\), \(\sigma^2\)), which gives the likelihood [black line]. Note that the likelihood is scaled so it fits nicely on the graph (remember, likelihoods only matter up to a constant, so you can scale them however is convenient).
Because the normal distribution is the conjugate prior for normal sampling, the posterior distribution is also a normal distribution, and is shown in red.
By Bayes theorem:
\[ \text{Pr}(\mu \, | \, \mathbf{y}, \sigma^2) \propto \text{Pr}(\mathbf{y} \, | \, \mu, \sigma^2) \text{Pr}(\mu) \]
\[ \text{N}(\mu_1, \tau_1^2) = \text{N}(\mu, \sigma^2) \text{N}(\mu_0, \tau_0^2) \]
where the posterior mean:
\[ \mu_1 = \frac{\frac{\mu_0}{\tau_0^2} + \frac{n \bar{y}}{\sigma^2}}{\frac{1}{\tau_0^2} + \frac{n}{\sigma^2}} \]
and the posterior variance:
\[ \tau_1^2 = (\frac{1}{\tau_0^2} + \frac{n}{\sigma^2})^{-1} \]
An interactive app that shows how the posterior distribution will change when the prior and the (scaled) data likelihood changes:
The source code of the app can be found here.
sessionInfo()R version 3.3.2 (2016-10-31)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 14.04.5 LTS
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
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] knitr_1.15.1 MASS_7.3-45 expm_0.999-0
[4] Matrix_1.2-8 workflowr_0.4.0 rmarkdown_1.3.9004
loaded via a namespace (and not attached):
[1] Rcpp_0.12.9 lattice_0.20-34 gtools_3.5.0 digest_0.6.12
[5] rprojroot_1.2 mime_0.5 R6_2.2.0 grid_3.3.2
[9] xtable_1.8-2 backports_1.0.5 git2r_0.18.0 magrittr_1.5
[13] evaluate_0.10 stringi_1.1.2 tools_3.3.2 stringr_1.2.0
[17] shiny_1.0.0 httpuv_1.3.3 yaml_2.1.14 htmltools_0.3.5This site was created with R Markdown