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This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.

This is incredibly artificial but to demonstrate the point let's say we wanted to fit the following model + priors:

$$ \begin{align} X_{ij} &\sim N(\mu_j, \sigma) \\ \mu_j &\sim \text{LogNormal}(\mu, \tau) \\ \mu &\sim \text{N}(2, 1) \\ \sigma &\sim \text{LogNormal}(0.5, 0.5) \\ \tau &\sim \text{LogNormal}(0.5, 0.5) \\ \end{align} $$

Now lets say our "question of interest" that we want to answer is "what is the population-mean of the $\mu_j$'s ?".

Naively I would assume that because they follow a Log-Normal distribution that the mean of the distribution is then $exp(\mu + \tau^2/2)$ (wiki) where we just sub in the posterior samples of $\mu$ and $\tau$ to get an uncertainty distribution.

But this doesn't feel right to me...

For example the posterior of $\tau$ is not necesarily a Log-Normal distribution anymore (that was just our prior), so why would the posterior population distribution of $\mu_j$ still be a Log-Normal distribution ? But if its not a Log-Normal distribution what do the values of $\mu$ and $\tau$ even mean anymore then ?

Any guidance as to where my mental model has gone wrong would be greatly appreciated.

This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.

This is incredibly artificial but to demonstrate the point let's say we wanted to fit the following model + priors:

$$ \begin{align} X_{ij} &\sim N(\mu_j, \sigma) \\ \mu_j &\sim \text{LogNormal}(\mu, \tau) \\ \mu &\sim \text{N}(2, 1) \\ \sigma &\sim \text{LogNormal}(0.5, 0.5) \\ \tau &\sim \text{LogNormal}(0.5, 0.5) \\ \end{align} $$

Now lets say our "question of interest" that we want to answer is "what is the population-mean of the $\mu_j$'s ?".

Naively I would assume that because they follow a Log-Normal distribution that the mean of the distribution is then $exp(\mu + \tau^2/2)$ (wiki) where we just sub in the posterior samples of $\mu$ and $\tau$ to get an uncertainty distribution.

But this doesn't feel right to me...

For example the posterior of $\tau$ is not necesarily a Log-Normal distribution anymore (that was just our prior), so why would the posterior population distribution of $\mu_j$ still be a Log-Normal distribution ? But if its not a Log-Normal distribution what do the values of $\mu$ and $\tau$ even mean anymore then; do they even have an interpretation if the underlying distributions are different in the posterior ?

Any guidance as to where my mental model has gone wrong would be greatly appreciated.

This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.

This is incredibly artificial but to demonstrate the point let's say we wanted to fit the following model + priors:

$$ \begin{align} X_{ij} &\sim N(\mu_j, \sigma) \\ \mu_j &\sim \text{LogNormal}(\mu, \tau) \\ \mu &\sim \text{N}(2, 1) \\ \sigma &\sim \text{LogNormal}(0.5, 0.5) \\ \tau &\sim \text{LogNormal}(0.5, 0.5) \\ \end{align} $$

Now lets say a question we want to answer is "what is the population-mean of the $\mu_j$'s ?".

Naively I would assume that because they follow a Log-Normal distribution that the mean of the distribution is then $exp(\mu + \tau^2/2)$ (wiki) where we just sub in the posterior samples of $\mu$ and $\tau$ to get an uncertainty distribution.

But this doesn't feel right to me...

For example the posterior of $\tau$ is not necesarily a Log-Normal distribution anymore (that was just our prior), so why would the posterior population distribution of $\mu_j$ still be a Log-Normal distribution ? But if its not a Log-Normal distribution what do the values of $\mu$ and $\tau$ even mean anymore then ?

Any guidance as to where my mental model has gone wrong would be greatly appreciated.

This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.

This is incredibly artificial but to demonstrate the point let's say we wanted to fit the following model + priors:

$$ \begin{align} X_{ij} &\sim N(\mu_j, \sigma) \\ \mu_j &\sim \text{LogNormal}(\mu, \tau) \\ \mu &\sim \text{N}(2, 1) \\ \sigma &\sim \text{LogNormal}(0.5, 0.5) \\ \tau &\sim \text{LogNormal}(0.5, 0.5) \\ \end{align} $$

Now lets say our "question of interest" that we want to answer is "what is the population-mean of the $\mu_j$'s ?".

Naively I would assume that because they follow a Log-Normal distribution that the mean of the distribution is then $exp(\mu + \tau^2/2)$ (wiki) where we just sub in the posterior samples of $\mu$ and $\tau$ to get an uncertainty distribution.

But this doesn't feel right to me...

For example the posterior of $\tau$ is not necesarily a Log-Normal distribution anymore (that was just our prior), so why would the posterior population distribution of $\mu_j$ still be a Log-Normal distribution ? But if its not a Log-Normal distribution what do the values of $\mu$ and $\tau$ even mean anymore then ?

Any guidance as to where my mental model has gone wrong would be greatly appreciated.

This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.

This is incredibly artificial but to demonstrate the point let's say we wanted to fit the following model + priors:

$$ \begin{align} X_{ij} &\sim N(\mu_j, \sigma) \\ \mu_j &\sim \text{LogNormal}(\mu, \tau) \\ \mu &\sim \text{N}(2, 1) \\ \sigma &\sim \text{LogNormal}(0.5, 0.5) \\ \tau &\sim \text{LogNormal}(0.5, 0.5) \\ \end{align} $$

Now lets say a question we want to answer is "what is the population-mean of the $\mu_j$'s ?".

Naively I would assume that because they follow a Log-Normal distribution that the mean of the distribution is then $exp(\mu + \tau^2/2)$ (wiki) where we just sub in the posterior estimates of $\mu$ and $\tau$.

But this doesn't feel right to me...

For example the posterior of $\tau$ is not necesarily a Log-Normal distribution anymore (that was just our prior), so why would the posterior population distribution of $\mu_j$ still be a Log-Normal distribution ? But if its not a Log-Normal distribution what do the values of $\mu$ and $\tau$ even mean anymore then ?

Any guidance as to where my mental model has gone wrong would be greatly appreciated.

This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.

This is incredibly artificial but to demonstrate the point let's say we wanted to fit the following model + priors:

$$ \begin{align} X_{ij} &\sim N(\mu_j, \sigma) \\ \mu_j &\sim \text{LogNormal}(\mu, \tau) \\ \mu &\sim \text{N}(2, 1) \\ \sigma &\sim \text{LogNormal}(0.5, 0.5) \\ \tau &\sim \text{LogNormal}(0.5, 0.5) \\ \end{align} $$

Now lets say a question we want to answer is "what is the population-mean of the $\mu_j$'s ?".

Naively I would assume that because they follow a Log-Normal distribution that the mean of the distribution is then $exp(\mu + \tau^2/2)$ (wiki) where we just sub in the posterior samples of $\mu$ and $\tau$ to get an uncertainty distribution.

But this doesn't feel right to me...

For example the posterior of $\tau$ is not necesarily a Log-Normal distribution anymore (that was just our prior), so why would the posterior population distribution of $\mu_j$ still be a Log-Normal distribution ? But if its not a Log-Normal distribution what do the values of $\mu$ and $\tau$ even mean anymore then ?

Any guidance as to where my mental model has gone wrong would be greatly appreciated.