When scientists are interested in knowing the values of, or inferring causal relations between variables that they cannot directly measure, they typically record several survey or test ?items? that are thought to be indicators of latent variables of interest, e.g. math ability or impulsiveness. Although it is rare that a latent variable is measured perfectly by any single indicator, estimates of the latent variables, and their associations with other latents, can be obtained by employing multiple indicators for each latent variable (multiple indicator models). If the multiple indicator model is correctly specified, then in a wide range of cases the estimates obtained are consistent and have desirable statistical properties, and these estimates can be used to search for causal models among the latents. If the model is mis-specified, then estimators are typically biased.

A number of problems make it difficult to find a correctly specified measurement model: a) associations among items are often confounded by additional unknown latent common causes, b) there are often a plethora of alternative models that are consistent with the data and with the prior knowledge of domain experts, c) there may be non-linear dependencies among latent variables, or linear relationships among non-Gaussian latent variables, and d) there may be feedback relationships among latent variables. This work will generalize previous work by Silva, et al., 2006 in JMLR in a way that overcomes many of these difficulties. In particular, I will sketch an outline to an approach that leverages algebraic work by Sullivant et al. (2010) in the Annals of Statistics that allows for models with several latent factors underlying each pair of items, that involve non-linearities and non-Gaussian variables (in parts of the model), and that involve feedback between latent variables of interest; I will also describe what the open problems for this approach are.