Joint "density" of data and indicators in Bayesian mixture model I'm currently working through the chapter on finite mixture models in
BDA3 and came across the
following model setup (with the usual slight abuse of notation):
Let $\lambda=(\lambda_1,\dots,\lambda_H)$, $\sum_{h=1}^H\lambda_h = 1$ be the
mixing proportions and let $f(y_i\mid\theta_h)$ be some parametric density.
Then the observations $y=(y_1,\dots,y_n)$ can be modeled as a continuous real-valued random variable with density
$$p(y\mid \theta,\lambda) = \prod_{i=1}^n\sum_{h=1}^H \lambda_h f(y_i\mid\theta_h).$$
Alternatively we can introduce latent class indicators $z_{ih}$
$$z_{ih} = \begin{cases}1 & \text{if i-th observation is drawn from h-th mixture component} \\ 0 &\text{otherwise}\end{cases} $$
with distribution
$$z_i = (z_{i1},\dots,z_{iH}) \mid \lambda \sim \text{Multinomial}(1; \lambda)$$
and corresponding probability mass function $p(z_i\mid\lambda)$ which lets us write
$$p(y\mid z, \theta) = \prod_{i=1}^n \prod_{h=1}^H (f(y_i\mid\theta_h))^{z_{ih}}.$$
So far things seem clear. But then the authors write:

The joint distribution of the observed data $y$ and the unobserved indicators $z$ conditional on the model parameters can be written
  $$p(y, z\mid\theta,\lambda) = p(z\mid\lambda)p(y\mid z, \theta) = \prod_{i=1}^n\prod_{h=1}^H(\lambda_hf(y_i\mid\theta_h))^{z_{ih}}$$
  with exactly one of $z_{ih}$ equaling $1$ for each $i$.

From the perspective of probability theory we can not just multiply
$p(y\mid z,\theta)$ and $p(z\mid\lambda)$, as the former is the pdf of a
continous random variable and the latter is the pmf of a discrete random
variable. On the other hand, we can multiply with priors for $\lambda$
and $\theta$, work out the full conditionals and get a working Gibbs
sampler.
So, my question is: What is $p(y, z\mid\theta,\lambda)$ as defined above, if it
is neither a pdf nor pmf?
 A: So, one year and a thorough course on probability later this seems kind of
obvious, but maybe the following sketch is still helpful for someone:


*

*Probability density functions and probability mass functions are the same
kind of object. Both are a Radon-Nikodym
derivative of a
measure; in the case of the pdf with regard to ($n$-dimensional)
Lebesgue-Measure, in the case of the pmf with regard to a counting measure.

*The general case of Bayes' theorem relates the prior distribution to the
Markov kernel that is the
posterior "distribution" by way of a Markov kernel for the conditional distribution of
the observables given the parameters. This need not involve any density or
likelihood functions at all.

*The densities for the prior distribution and for the conditional distribution
of observables given parameters can very well be taken with regard to
different dominating measures. In that case the joint distribution of
observations and parameters has a density with regard to the product
measure of the dominating
measures.

*In my case this means $p(y, z\mid\theta,\lambda)$ is a density of the conditional
distribution of $y$ and $z$ given $\theta$ and $\lambda$ with regard to the product of Lebesgue measure
and a counting measure on $\{1,\dots,H\}$.

