Bayesian inference data from independent normal two different prior assumptions

Question

Say we have $n$ data points $x_1, x_2, ..., x_n$, let's also assume that each data point comes from a normal distribution. For Bayesian inference,

the first prior assumption is, for each data point $x_i$, $i=1, ..., n$, we assume $x_i \sim N(\mu_i,\sigma^2)$, also assume that $\sigma^2$ is known, so we only need to specify a prior on each $\mu_i$, we assume $\mu_i \sim N(\lambda_1,\lambda_2)$;

the second prior assumption is, for each data point $x_i$, $i=1, ..., n$, we assume $x_i \sim N(\mu,\sigma^2)$, $\sigma^2$ is known, we also assume $\mu \sim N(\lambda_1,\lambda_2)$;

apparently, if we do Bayesian inference for the posterior distribution. The two prior assumptions will give us two different results, for the first one, we have $n$ posterior distributions for each $\mu_i$, and for the second one, we only have one posterior distribution for $\mu$. However, what are the differences between the two prior assumptions, it seems like they are the same. And which one is more appropriate?

Answer 1

The two assumptions are not identical. By using the theorem of total expectation (and of total variance) we use the assumption 1. to calculate the marginal distribution for $x_i$. It is normal, with expectation and variance given by: $$\DeclareMathOperator{\E}{\mathbb{E}} \DeclareMathOperator{\V}{\mathbb{V}} \E x_i = \E \E x_i \mid \mu_i = \E \mu = \lambda_1 \\ \V x_i = \E \V x_i \mid \mu_i + \V \E x_i \mid \mu_i = \\ \E \sigma^2 + \V \lambda_1 = \sigma^2 + \lambda_1 $$ which is not the same as given under assumption 2.

Se also the comments under the question.

To get equality you must replace assumption 2. by $$ x_i \sim N(\mu_i,\sigma^2+\lambda_1) $$