# Theory of Bayesian Estimation

I'm starting with Bayesian analysis, and I've been trying to understand how to write down a Bayesian model. Let's suppose a one-way random effects ANOVA model:

$y_{ij}|\mu_i, \sigma^2 \sim N(\mu_i, \sigma^2)$

$\mu_i|\mu, D \sim N(\mu, D)$

$\mu \sim N(m, v)$

$\sigma^2 \sim InvGamma(a/2, b/2)$

$D \sim InvGamma(c/2, d/2)$

• Suppose I knew the grand mean $\mu$ were greater than 50? How can I change the assumptions of my model above to take this into account?

• Suppose the sampling variance $\sigma^2$ were between 6 and 11? How should I change the assumptions of my ANOVA model to take this into account?

• Let's suppose I'm doing an experiment on subjects $j$ with $n_j$ repeated observations $y_{ij}$ on each subject. I assign subjects to be in one of the two groups $i$: (a) treatment $i = 1, ..., I/2$, or (b) control $i = I/2 + 1, ..., I$. How should I change my ANOVA model to allow the population mean of subjects in the treatment group to be different from those in the control group?

Thanks!

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