I have a vector parameter $\vec{\theta} = [\theta_{1}, \theta_{2}, \theta_{3}]^{T}$, each of the component being independent random variable.
The data generating model is $p({\vec{h}} | \vec{\theta})$ with likelihood function $L(\vec{\theta} | \vec{h})$ and unninformative prior $\pi(\vec{\theta}) \propto \vec{1}$
Am I correct that the Bayesian posterior for the parameter vector, up to normalisation constant, can be expressed as
$p(\vec{\theta} | \vec{h}) \propto L(\vec{\theta} | \vec{h}) = L(\theta_{1}, \theta_{2}, \theta_{3} | \vec{h}) = L(\theta_{1} | \vec{h}).L(\theta_{2} | \vec{h}).L(\theta_{3} | \vec{h})$?
If not, where am I going wrong?