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