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EDIT - I want to add that I no longer think my answer below is correct (i'm still unsure as to what I think the proper answer is - my current suspicion is that the "question of interest" can't be answered in a Bayesian framework) but am leaving it up as it already has comments.

(posting an answer to my own question) I'm not sure if this is correct but after thinking about this for ages (and based on the answer by @LmnICE) the following seems to make some intuiative sense to me, please comment/correct me if I have this wrong

I think the confusion is over what the posterior actually represents. Being explicit our posterior as noted in the OP can be expressed as:

$$ P(\mu_j, \mu, \tau, \sigma \ \mid X) \propto P(X \mid \mu_j, \sigma)P(\mu_j \mid \mu, \tau)P(\tau)P(\sigma)P(\mu) $$

That is to say that a sample of $\mu_j, \mu, \tau, \sigma$ from our posterior distribution is equivalent to a sample from the model which assumes these relationships between the parameters.

Or put another way, the posterior hasn't changed any of these assumed relationships; in fact the posterior is dependent on them. E.g. a sample of $\mu$ from the posterior is equivalent to a sample of $\mu$ where we are assuming it is the log-centrality parameter of a log-normal distribution for $\mu_j$.

I think the confusion I had is that we often refer to the posterior as being an updated / changed distribution but this is merely referencing the fact that $P(\theta \mid X) \ne P(\theta)$. However under this model the data doesn't change the relationship between parameters e.g. $P(\mu_j \mid \mu, \tau)$ is still $P(\mu_j \mid \mu, \tau)$, all that is changed is what values we think those parameters are.

So to answer "question of interest" I would argue that as our model is assuming a log-normal releationship that the population-mean of $\mu_j$ is indeed $exp(\mu + \tau^2/2)$. Likewise then the interpretation of the values of $\mu$ and $\tau$ that come out of the posterior are the values that most likely fit this assumed model given the available data.

EDIT - I want to add that I no longer think my answer below is correct (i'm still unsure as to what I think the answer is) but am leaving my answer up as it already has comments.

(posting an answer to my own question) I'm not sure if this is correct but after thinking about this for ages (and based on the answer by @LmnICE) the following seems to make some intuiative sense to me, please comment/correct me if I have this wrong

I think the confusion is over what the posterior actually represents. Being explicit our posterior as noted in the OP can be expressed as:

$$ P(\mu_j, \mu, \tau, \sigma \ \mid X) \propto P(X \mid \mu_j, \sigma)P(\mu_j \mid \mu, \tau)P(\tau)P(\sigma)P(\mu) $$

That is to say that a sample of $\mu_j, \mu, \tau, \sigma$ from our posterior distribution is equivalent to a sample from the model which assumes these relationships between the parameters.

Or put another way, the posterior hasn't changed any of these assumed relationships; in fact the posterior is dependent on them. E.g. a sample of $\mu$ from the posterior is equivalent to a sample of $\mu$ where we are assuming it is the log-centrality parameter of a log-normal distribution for $\mu_j$.

I think the confusion I had is that we often refer to the posterior as being an updated / changed distribution but this is merely referencing the fact that $P(\theta \mid X) \ne P(\theta)$. However under this model the data doesn't change the relationship between parameters e.g. $P(\mu_j \mid \mu, \tau)$ is still $P(\mu_j \mid \mu, \tau)$, all that is changed is what values we think those parameters are.

So to answer "question of interest" I would argue that as our model is assuming a log-normal releationship that the population-mean of $\mu_j$ is indeed $exp(\mu + \tau^2/2)$. Likewise then the interpretation of the values of $\mu$ and $\tau$ that come out of the posterior are the values that most likely fit this assumed model given the available data.

EDIT - I want to add that I no longer think my answer below is correct (i'm still unsure as to what I think the proper answer is) but am leaving it up as it already has comments.

(posting an answer to my own question) I'm not sure if this is correct but after thinking about this for ages (and based on the answer by @LmnICE) the following seems to make some intuiative sense to me, please comment/correct me if I have this wrong

I think the confusion is over what the posterior actually represents. Being explicit our posterior as noted in the OP can be expressed as:

$$ P(\mu_j, \mu, \tau, \sigma \ \mid X) \propto P(X \mid \mu_j, \sigma)P(\mu_j \mid \mu, \tau)P(\tau)P(\sigma)P(\mu) $$

That is to say that a sample of $\mu_j, \mu, \tau, \sigma$ from our posterior distribution is equivalent to a sample from the model which assumes these relationships between the parameters.

Or put another way, the posterior hasn't changed any of these assumed relationships; in fact the posterior is dependent on them. E.g. a sample of $\mu$ from the posterior is equivalent to a sample of $\mu$ where we are assuming it is the log-centrality parameter of a log-normal distribution for $\mu_j$.

I think the confusion I had is that we often refer to the posterior as being an updated / changed distribution but this is merely referencing the fact that $P(\theta \mid X) \ne P(\theta)$. However under this model the data doesn't change the relationship between parameters e.g. $P(\mu_j \mid \mu, \tau)$ is still $P(\mu_j \mid \mu, \tau)$, all that is changed is what values we think those parameters are.

So to answer "question of interest" I would argue that as our model is assuming a log-normal releationship that the population-mean of $\mu_j$ is indeed $exp(\mu + \tau^2/2)$. Likewise then the interpretation of the values of $\mu$ and $\tau$ that come out of the posterior are the values that most likely fit this assumed model given the available data.

EDIT - I want to add that I no longer think my answer below is correct (i'm still unsure as to what I think the proper answer is - my current suspicion is that the "question of interest" can't be answered in a Bayesian framework) but am leaving it up as it already has comments.

(posting an answer to my own question) I'm not sure if this is correct but after thinking about this for ages (and based on the answer by @LmnICE) the following seems to make some intuiative sense to me, please comment/correct me if I have this wrong

I think the confusion is over what the posterior actually represents. Being explicit our posterior as noted in the OP can be expressed as:

$$ P(\mu_j, \mu, \tau, \sigma \ \mid X) \propto P(X \mid \mu_j, \sigma)P(\mu_j \mid \mu, \tau)P(\tau)P(\sigma)P(\mu) $$

That is to say that a sample of $\mu_j, \mu, \tau, \sigma$ from our posterior distribution is equivalent to a sample from the model which assumes these relationships between the parameters.

Or put another way, the posterior hasn't changed any of these assumed relationships; in fact the posterior is dependent on them. E.g. a sample of $\mu$ from the posterior is equivalent to a sample of $\mu$ where we are assuming it is the log-centrality parameter of a log-normal distribution for $\mu_j$.

I think the confusion I had is that we often refer to the posterior as being an updated / changed distribution but this is merely referencing the fact that $P(\theta \mid X) \ne P(\theta)$. However under this model the data doesn't change the relationship between parameters e.g. $P(\mu_j \mid \mu, \tau)$ is still $P(\mu_j \mid \mu, \tau)$, all that is changed is what values we think those parameters are.

So to answer "question of interest" I would argue that as our model is assuming a log-normal releationship that the population-mean of $\mu_j$ is indeed $exp(\mu + \tau^2/2)$. Likewise then the interpretation of the values of $\mu$ and $\tau$ that come out of the posterior are the values that most likely fit this assumed model given the available data.

(posting an answer to my own question) I'm not sure if this is correct but after thinking about this for ages (and based on the answer by @LmnICE) the following seems to make some intuiative sense to me, please comment/correct me if I have this wrong

I think the confusion is over what the posterior actually represents. Being explicit our posterior as noted in the OP can be expressed as:

$$ P(\mu_j, \mu, \tau, \sigma \ \mid X) \propto P(X \mid \mu_j, \sigma)P(\mu_j \mid \mu, \tau)P(\tau)P(\sigma)P(\mu) $$

That is to say that a sample of $\mu_j, \mu, \tau, \sigma$ from our posterior distribution is equivalent to a sample from the model which assumes these relationships between the parameters.

Or put another way, the posterior hasn't changed any of these assumed relationships; in fact the posterior is dependent on them. E.g. a sample of $\mu$ from the posterior is equivalent to a sample of $\mu$ where we are assuming it is the log-centrality parameter of a log-normal distribution for $\mu_j$.

I think the confusion I had is that we often refer to the posterior as being an updated / changed distribution but this is merely referencing the fact that $P(\theta \mid X) \ne P(\theta)$. However under this model the data doesn't change the relationship between parameters e.g. $P(\mu_j \mid \mu, \tau)$ is still $P(\mu_j \mid \mu, \tau)$, all that is changed is what values we think those parameters are.

So to answer "question of interest" I would argue that as our model is assuming a log-normal releationship that the population-mean of $\mu_j$ is indeed $exp(\mu + \tau^2/2)$. Likewise then the interpretation of the values of $\mu$ and $\tau$ that come out of the posterior are the values that most likely fit this assumed model given the available data.

EDIT - I want to add that I no longer think my answer below is correct (i'm still unsure as to what I think the proper answer is) but am leaving it up as it already has comments.

(posting an answer to my own question) I'm not sure if this is correct but after thinking about this for ages (and based on the answer by @LmnICE) the following seems to make some intuiative sense to me, please comment/correct me if I have this wrong

I think the confusion is over what the posterior actually represents. Being explicit our posterior as noted in the OP can be expressed as:

$$ P(\mu_j, \mu, \tau, \sigma \ \mid X) \propto P(X \mid \mu_j, \sigma)P(\mu_j \mid \mu, \tau)P(\tau)P(\sigma)P(\mu) $$

That is to say that a sample of $\mu_j, \mu, \tau, \sigma$ from our posterior distribution is equivalent to a sample from the model which assumes these relationships between the parameters.

Or put another way, the posterior hasn't changed any of these assumed relationships; in fact the posterior is dependent on them. E.g. a sample of $\mu$ from the posterior is equivalent to a sample of $\mu$ where we are assuming it is the log-centrality parameter of a log-normal distribution for $\mu_j$.

I think the confusion I had is that we often refer to the posterior as being an updated / changed distribution but this is merely referencing the fact that $P(\theta \mid X) \ne P(\theta)$. However under this model the data doesn't change the relationship between parameters e.g. $P(\mu_j \mid \mu, \tau)$ is still $P(\mu_j \mid \mu, \tau)$, all that is changed is what values we think those parameters are.

So to answer "question of interest" I would argue that as our model is assuming a log-normal releationship that the population-mean of $\mu_j$ is indeed $exp(\mu + \tau^2/2)$. Likewise then the interpretation of the values of $\mu$ and $\tau$ that come out of the posterior are the values that most likely fit this assumed model given the available data.