In this letter, we present an improved method for the independence test procedure in the convolutional multicast algorithm proposed by Erez and Feder. We employ the linear independence test vectors to check the independence of the partial encoding vectors in the main program of Erez's convolutional multicast algorithm. It turns out that compared with the previous approach of computing the determinants of the correlative matrices, carrying out the independence test vectors can reduce the computational complexity.

Squared-loss mutual information (SMI) is a robust measure of the statistical dependence between random variables. The sample-based SMI approximator called least-squares mutual information (LSMI) was demonstrated to be useful in performing various machine learning tasks such as dimension reduction, clustering, and causal inference. The original LSMI approximates the pointwise mutual information by using the kernel model, which is a linear combination of kernel basis functions located on paired data samples. Although LSMI was proved to achieve the optimal approximation accuracy asymptotically, its approximation capability is limited when the sample size is small due to an insufficient number of kernel basis functions. Increasing the number of kernel basis functions can mitigate this weakness, but a naive implementation of this idea significantly increases the computation costs. In this article, we show that the computational complexity of LSMI with the multiplicative kernel model, which locates kernel basis functions on unpaired data samples and thus the number of kernel basis functions is the sample size squared, is the same as that for the plain kernel model. We experimentally demonstrate that LSMI with the multiplicative kernel model is more accurate than that with plain kernel models in small sample cases, with only mild increase in computation time.

Identifying the statistical independence of random variables is one of the important tasks in statistical data analysis. In this paper, we propose a novel non-parametric independence test based on a least-squares density ratio estimator. Our method, called least-squares independence test (LSIT), is distribution-free, and thus it is more flexible than parametric approaches. Furthermore, it is equipped with a model selection procedure based on cross-validation. This is a significant advantage over existing non-parametric approaches which often require manual parameter tuning. The usefulness of the proposed method is shown through numerical experiments.

Item response theory (IRT) is widely used for test analyses. Most models of IRT assume that a subject's responses to different items in a test are statistically independent. However, actual situations often violate this assumption. Thus, conditional independence (CI) tests among items given a latent ability variable are needed, but traditional CI tests suffer from biases. This study investigated a latent conditional independence (LCI) test given a latent variable. Results show that the LCI test can detect CI given a latent variable correctly, whereas traditional CI tests often fail to detect CI. Application of the LCI test to mathematics test data revealed that items that share common alternatives might be conditionally dependent.