Parameters not empirically identifiable or distinguishable, including correlation between Gaussian observations

2023/03/05 by Christian Hennig · 1 voice
Computer Science · Mathematics · #Bayesian Modeling and Causal Inference #Bayesian Methods and Mixture Models #Statistical Methods and Inference #Identifiability #Gaussian #Estimator #Correlation #Independence (probability theory) #Mathematics #Statistical physics #Constant (computer programming) #Econometrics #Cluster analysis #Statistics #Computer science #Physics

paper · doi:10.1007/s00362-023-01414-3

openalex publication_date 2023/03/05 · openalex created_date 2023/03/06 · openalex updated_date 2026/08/01

Abstract

Abstract It is shown that some theoretically identifiable parameters cannot be empirically identified, meaning that no consistent estimator of them can exist. An important example is a constant correlation between Gaussian observations (in presence of such correlation not even the mean can be empirically identified). Empirical identifiability and three versions of empirical distinguishability are defined. Two different constant correlations between Gaussian observations cannot even be empirically distinguished. A further example are cluster membership parameters in k -means clustering. Several existing results in the literature are connected to the new framework. General conditions are discussed under which independence can be distinguished from dependence.

Citations

• A General Structural Equation Model with Dichotomous, Ordered Categorical, and Continuous Latent Variable Indicators
• Data clustering: 50 years beyond K-means
• Identification in Parametric Models
• On a Test Whether Two Samples are from the Same Population
• On the Identifiability of Finite Mixtures

Discussions

• Not sure whether this is allowed in an answer, but I think it's useful to cite myself from an answer I wrote to a somewhat related (but not identical) question on philosophy.stackexchange: How can one [stackexchange, 1 points]