I'm learning Bayesian statistics and want to verify my understanding of a few things.
Let's say my data $X$ follows the model $f(x \mid \theta)$ with a prior $\pi(\theta)$. After observations $\mathbf{x} = x_1,\ldots,x_n$, the posterior distribution $\pi(\theta \mid \mathbf{x})$ is proportonal to the likelihood times the prior; i.e.,
$$\pi(\theta \mid \mathbf{x}) \propto L(\mathbf{x} \mid \theta) \cdot \pi(\theta).$$
where $$L(\mathbf{x} \mid \theta) = \prod_{i = 1}^n f(x_i \mid \theta).$$
I feel fairly confident about this.
Now let's suppose that my data $X$ follows the model $f(x \mid \theta_1, \theta_2)$ with independent and identically distributed priors $\pi(\theta_1)$ and $\pi(\theta_2)$. Again, let's say we have observations $\mathbf{x} = x_1,\ldots,x_n$.
Could you please confirm my current understanding of the following? (Or explain why my understanding is faulty?)
Understanding 1: Is the posterior distribution proportional to likelihood times both priors? i.e.
$$\pi(\theta_1, \theta_2 \mid \mathbf{x}) \propto L(\mathbf{x} \mid \theta_1, \theta_2) \cdot \pi(\theta_1) \cdot \pi(\theta_2).$$
Understanding 2: Is the conditional posterior distribution of $\theta_1$ proportional to $L(\mathbf{x} \mid \theta_1, \theta_2) \cdot \pi(\theta_1)$?
Understanding 3: Is the marginal posterior distribution of $\theta_1$ proportional to $\displaystyle\int_{\Theta_2} L(\mathbf{x} \mid \theta_1, \theta_2) \cdot \pi(\theta_1) \cdot \pi(\theta_2) \,\mathrm{d}\theta_2$ ?
Much appreciated!
Yes.
You could see the parameters like a vector: $\theta = (\theta_1, \theta_2)$ where the prior for $\theta$ is given by the joint of the priors of the two components above, but since they are independent it is just the product: $\pi(\theta) = \pi(\theta_1)\pi(\theta_2)$. Thus your posterior follows from this consideration.
This is just the chain rule for conditioning, so indeed it is proportional to you what you wrote.
The third one also correct, since by integrating out over $\Theta_2$ you get indeed that the expression you gave is proportional to the posterior of $\theta_1$