What is the difference between auxiliary variable and Latent variable when we talk about joint posterior density (when it is in complex form). Notice that, my work is based on continuous random variables. I want an example to understand it.
Basically, an auxiliary variable is a hyper-parameter without any direct interpretation which is introduced for technical/simulation reasons or for the reason of making an analytically intractable distribution tractable. For example, when parameterising the student's t distribution you may introduce a $\chi^2$ distributed auxiliary variance modification parameter into a normal distribution with mean $\mu$ and precision $\lambda$ $$ y \sim N(\mu, 1/(\lambda \cdot s))$$ $$ s \sim \chi^2(\nu/2)$$ which adjusts the lambda on an additional hierarchical layer such that the resulting distribution for y will be a non-central student's t with $\nu$ degrees of freedom. A proof and detailed formula description of this relation you can look up for example in this post. Note that s has no interpretation in this model whatsoever; this is just an auxiliary parameter to help us simulate and update the t-distribution more easily. A latent variable is a variable which has an interpretation, but cannot be observed directly. Hidden Markov models typically deal with these. Imagine in a biological system you observe how a plant is growing. Yet the height of this plant is not what you are interested in, but it's genetic acitivites which are hidden as you cannot measure them, but model them on a 'hidden' layer between your observations and the technical/auxiliary parameters. As long as you do not wish to interpret this latent parameter, it is still an auxiliary variable.