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.

Session information

R version 3.3.2 (2016-10-31)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 14.04.5 LTS

 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=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_0.3.5

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