5.13 Normal-Normal conjugate family

Let’s start considering a case study.

Among all people who have a history of concussions, we are interested in $\mu$ the average volume (in cubic centimeters) of a specific part of the brain: the hippocampus.

What we know so far:

• Both halves of the adults’ hippocampus have a volume between 3.0 and 3.5 $cm^3$.

• Total hippocampal volume of both sides of the brain is between 6 and 7 $cm^3$

Our starting hypotheses are:

• $H_0$: $\mu$ is between 6 and 7 $cm^3$ with an average of 6.5.
• $n=25$
• suppose $Normal$ prior
• use the $Normal-Normal$ Bayesian model

Since hippocampal volumes, our $Y_i$, are measured on a continuous scale, there are many possible common models of the variability:

• Beta –> but it is not good as it assumes $Y_i \epsilon[0,1]$, our $\mu$ is 6.5$cm^3$
• Exponential
• Gamma
• Normal –> hippocampal symmetrically or Normally distributed around some global average
• F
• …etc