Last updated: 2017-03-04

Code version: 5d0fa13282db4a97dc7d62e2d704e88a5afdb824

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.

Session information

R version 3.3.0 (2016-05-03)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X 10.10.5 (Yosemite)

[1] en_NZ.UTF-8/en_NZ.UTF-8/en_NZ.UTF-8/C/en_NZ.UTF-8/en_NZ.UTF-8

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

other attached packages:
 [1] tidyr_0.4.1     dplyr_0.5.0     ggplot2_2.1.0   knitr_1.15.1   
 [5] MASS_7.3-45     expm_0.999-0    Matrix_1.2-6    viridis_0.3.4  
 [9] workflowr_0.3.0 rmarkdown_1.3  

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.5      git2r_0.18.0     plyr_1.8.4       tools_3.3.0     
 [5] digest_0.6.9     evaluate_0.10    tibble_1.1       gtable_0.2.0    
 [9] lattice_0.20-33  shiny_0.13.2     DBI_0.4-1        yaml_2.1.14     
[13] gridExtra_2.2.1  stringr_1.2.0    gtools_3.5.0     rprojroot_1.2   
[17] grid_3.3.0       R6_2.1.2         reshape2_1.4.1   magrittr_1.5    
[21] backports_1.0.5  scales_0.4.0     htmltools_0.3.5  assertthat_0.1  
[25] mime_0.5         colorspace_1.2-6 xtable_1.8-2     httpuv_1.3.3    
[29] labeling_0.3     stringi_1.1.2    munsell_0.4.3   

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