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Missing Not At Random (MNAR) values lead to significant biases in the data,since the probability of missingness depends on the unobserved values.They are''not ignorable'' in the sense that they often require defining a model for themissing data mechanism, which makes inference or imputation tasks more complex.Furthermore, this implies a strong \textit{a priori} on the parametric form ofthe distribution.However, some works have obtained guarantees on the estimationof parameters in the presence of MNAR data, without specifying the distributionof missing data \citep{mohan2018estimation, tang2003analysis}. This is veryuseful in practice, but is limited to simple cases such as self-masked MNARvalues in data generated according to linear regression models.We continue thisline of research, but extend it to a more general MNAR mechanism, in a moregeneral model of the probabilistic principal component analysis (PPCA),\textit{i.e.}, a low-rank model with random effects. We prove identifiabilityof the PPCA parameters. We then propose an estimation of the loadingcoefficients and a data imputation method. They are based on estimators ofmeans, variances and covariances of missing variables, for which consistency isdiscussed. These estimators have the great advantage of being calculated usingonly the observed data, leveraging the underlying low-rank structure of thedata. We illustrate the relevance of the method with numerical experiments onsynthetic data and also on real data collected from a medical register.

Estimation and imputation in Probabilistic Principal Component Analysis with Missing Not At Random data