# Bayes version for continuous case, what does the integral mean?

In bayes version for continuous case, what does it mean to integrate with respect to $$d\theta$$ when $$\theta$$ is a vector not a a scalar value?

$$p(\theta|D) = \frac{p(D|\theta)p(\theta)}{p(D)}$$

Where $$D$$ is a set of observed data points, and $$\theta$$ is a vector of parameters to be estimated.

$$p(D) = \int p(D|\theta)p(\theta)d\theta$$

Is this integral a surface integral, component-wise integral or what ?

• You are computing the marginal distribution of $p(D)$ over the range of your parameter space $\theta \in \Theta$. The choice of integral depends on how your parameter space is defined. – Maxtron Sep 22 '18 at 16:48
• Can you give an example with a specific parameter space? I'm not sure I fully understand your answer – Loai Ghoraba Sep 22 '18 at 17:27
• Say your parameter $\theta$ is a 2-D vector and bounded, i.e., $\theta_1 \in [a_1, b_1]$ and $\theta_2 \in [a_2, b_2]$, then $\Theta \in \{ [a_1, b_1] \times [a_2, b_2] \}$, which is a surface. – Maxtron Sep 22 '18 at 18:01

That will be the n-dimensional integral, like the 2D case below: $$p(D) = \int_\Theta\int p(D | \theta_1,\theta_2)p(\theta_1,\theta_2)d\theta_1d\theta_2$$ Because it does not matter if you put them in a vector or not; if you formulate your likelihood and joint PDF correctly. You just have $$n$$ random variables.
• Just to make sure, so it is not a surface integral, right? That is , $d\theta$ is just a compact representation of multiple deltas multiplied together ? – Loai Ghoraba Sep 22 '18 at 21:03
• Well, what would you call integrals with $dxdy$? – gunes Sep 22 '18 at 21:12