# Mixture modelling of data with measurement uncertainty

I have a dataset that consists of a population radiometric ages (300>n>600). A dataset can have ages can range on the order of billions of years. Each age measurement has an associated uncertainty based on our precision of measuring isotopic ratios. The problem I am trying to solve is how to identify the youngest sub-population of ages in this group that is separate from another, older, sub-populations.

Most mixture models work under the assumptions that each data point has no uncertainty. I believe that for my problem that including the uncertainty is key. I am having a hard time finding an algorithm that incorporates measurement uncertainty into mixture models.

Additionally, in our process for deriving these radiometric ages we use reference materials to calibrate and validate our methods. We re-date a single reference material multiple times per analytical session and as such, we can understand the uncertainty of our method, beyond the calculated uncertainty of each datum. The incorporation of these data to help set the size of the clusters may be very powerful.

I was wondering if anyone has tried to solve a problem similar to mine and has any suggestions of approaches, literature, or algorithms that may be of use.

Consider the "base case" where $x$ is a collection of observations and $\theta$ is a parameter vector. The joint distribution is given by $$p(x,\theta) = p(x|\theta)\,p(\theta),$$ where $p(x|\theta)$ is likelihood (once $x$ is observed) and $p(\theta)$ is the prior distribution for $\theta$. The posterior distribution for $\theta$ can be expressed as $$p(\theta|x) \propto p(x|\theta)\,p(\theta) .$$
Now suppose that $x$ is not observed directly, but only indirectly via $y$ due to measurement error (for example). The joint distribution is now given by $$p(x,y,\theta) = p(y|x)\,p(x|\theta)\,p(\theta).$$ The posterior distribution for the unobserved entities (variable and parameters) is $$p(x,\theta|y) \propto p(y|x)\, p(x|\theta)\,p(\theta) .$$ The marginal posterior for $\theta$ is given by $$p(\theta|y) = \int p(x,\theta|y)\,dx .$$
The joint posterior distribution for $x$ and $\theta$ can be characterized by the following full conditional distributions: \begin{align} p(\theta|x,y) &\propto p(x|\theta)\,p(\theta) \\ p(x|y,\theta) &\propto p(y|x)\,p(x|\theta) . \end{align} The righthand sides incorporate important simplifications. First, the conditional distribution for $\theta$ does not depend on $y$. Second, the conditional distribution for $x$ does not depend on the prior for $\theta$. The Gibbs sampler makes draws from the joint posterior by alternating between the two conditional posteriors.