# Does this model simplification hold

I have a joint model where I have the output variable $Y$ and independent variable $X$ where I treat each of the output variable as being independent of each other. Additionally, there are model parameters $\beta$ and $W$. Now, I have $i$ independent observations and I write the joint log model as:

$$\log p(Y, \beta, W|X) = \sum_{i}^{N}\log P(y_i|x_i, w_i, \beta) + \log P(\beta) + \sum_{i}^{N}\log P(w_i)$$

What I am wondering is that considering that each of the $y_i$s are independent of each other and the same for $w_i$, can I write:

$$\log p(y_i, \beta, w_i|x_i) = \log P(y_i|x_i, w_i, \beta) + \log P(\beta) + \log P(w_i)$$

I am having this confusion as there is no sum over the prior distribution $P(\beta)$ of course, so could not convince myself this should be the case. I was wondering given this joint model with the independence assumptions that I made, whether I can write the joint model for each of the $i$ variables in any particular way?

After thinking about it, I think I can write the joint model for each $i$th term as:

$$\log p(y_i, \beta, w_i|x_i) = \log P(y_i|x_i, w_i, \beta) + \frac{\log P(\beta)}{N} + \log P(w_i)$$

The reason I say this is because I can write the joint model as:

$$p(Y, \beta, W|X) = P(\beta) \prod_{i}^{N} P(y_i|x_i, w_i, \beta) \prod_{i}^{N} P(w_i) = \prod_{i}^{N}P(y_i|x_i, w_i, \beta)P(w_i)\frac{P(\beta)}{N}$$

And now since $y_i$s and $w_i$ are independent, I think I can write everything as described above.

Can someone verify if this is correct or if I am on the wrong track?

Assuming what I understand from your independence hypothesis, the two first equations are good: $$p(y_i,\beta,w_i|x_i)=p(y_i|\beta,w_i,x_i) \cdot p(\beta,w_i | x_i) = p(y_i|\beta,w_i,x_i) \cdot p(\beta) \cdot p(w_i)$$ and $$p(Y,\beta,W|X)=\prod (p(y_i | \beta,W,X)) \cdot p(\beta,W,X) = \prod p(y_i | \beta,w_i,x_i) \cdot p(\beta) \cdot p(W)=\prod p(y_i | \beta,w_i,x_i) \cdot p(\beta) \cdot p(w_i)$$
Your last equation is good except the thrid equality as the $\frac{1}{N}$ factor should not be here (I guess that you messed with sum but $\beta \prod( \alpha_i)$ equals $\prod (\beta \alpha_i)$).