According to Theorem 3, what is the source of bias in the temporal MSE objective, and under what condition is the bias non-zero?

The bias in the temporal MSE objective originates from autocorrelation between labels, quantified by the partial correlation ρij. The bias term is non-zero if and only if such autocorrelation exists, i.e., ρij > 0.

Theorem 3 shows that the temporal MSE objective is biased relative to the practical negative log-likelihood when labels are autocorrelated. The bias is expressed as a function of partial correlations ρij between labels Yi and Yj given the input L. Specifically, the bias arises because the temporal MSE treats each future time step as independent, whereas the labels are actually conditionally dependent. The bias term is non-zero if and only if autocorrelation exists, meaning ρij > 0. Empirical analysis of the partial correlation matrix of labels confirms substantial off-diagonal values, demonstrating that label autocorrelation is present in time series data.