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Why is it necessary to place the distributional assumption on the errors, i.e.

$y_i = X\beta + \epsilon_{i}$, with $\epsilon_{i} \sim \mathcal{N}(0,\sigma^{2})$.

Why not write

$y_i = X\beta + \epsilon_{i}$, with $y_i \sim \mathcal{N}(\hat{y},\sigma^{2})$,

where in either case $\epsilon_i = y_i - \hat{y}$.
I've seen it stressed that the distributional assumptions are placed on the errors, not the data, but without explanation.

I'm not really understanding the difference between these two formulations. Some places I see distributional assumptions being placed on the data (Bayesian lit. it seems mostly), but most times the assumptions are placed on the errors.

When modelling, why would/should one choose to begin with assumptions on one or the other?

Why is it necessary to place the distributional assumption on the errors, i.e.

$y_i = X\beta + \epsilon_{i}$, with $\epsilon_{i} \sim \mathcal{N}(0,\sigma^{2})$.

Why not write

$y_i = X\beta + \epsilon_{i}$, with $y_i \sim \mathcal{N}(X\hat{\beta},\sigma^{2})$,

where in either case $\epsilon_i = y_i - \hat{y}$.
I've seen it stressed that the distributional assumptions are placed on the errors, not the data, but without explanation.

I'm not really understanding the difference between these two formulations. Some places I see distributional assumptions being placed on the data (Bayesian lit. it seems mostly), but most times the assumptions are placed on the errors.

When modelling, why would/should one choose to begin with assumptions on one or the other?

Why is it necessary to place the distributional assumption on the errors, i.e.

$y_i = X\beta + \epsilon_{i}$, with $\epsilon_{i} \sim \mathcal{N}(0,\sigma^{2})$.

Why not write,

$y_i = X\beta + \epsilon_{i}$, with $y_i \sim \mathcal{N}(\hat{y},\sigma^{2})$,

where in either case, $\epsilon_i = y_i - \hat{y}$. I've seen it stressed that the distributional assumptions are placed on the errors, not the data, but without explanation.

I'm not really understanding the difference between these two formulations. Some places I see distributional assumptions being placed on the data (bayesian lit. it seems mostly), but most times the assumptions are placed on the errors. When modeling, why would/should one choose to begin with assumptions on one or the other?

Why is it necessary to place the distributional assumption on the errors, i.e.

$y_i = X\beta + \epsilon_{i}$, with $\epsilon_{i} \sim \mathcal{N}(0,\sigma^{2})$.

Why not write

$y_i = X\beta + \epsilon_{i}$, with $y_i \sim \mathcal{N}(\hat{y},\sigma^{2})$,

where in either case $\epsilon_i = y_i - \hat{y}$.
I've seen it stressed that the distributional assumptions are placed on the errors, not the data, but without explanation.

I'm not really understanding the difference between these two formulations. Some places I see distributional assumptions being placed on the data (Bayesian lit. it seems mostly), but most times the assumptions are placed on the errors.

When modelling, why would/should one choose to begin with assumptions on one or the other?