The reconciliation of process measurements, subject to linear constraints, has usually involved finding the minimum weighted sum of squares of adjustments to the measurements. In order to do statistical tests, the data are most commonly assumed to follow a multivariate normal distribution. In this paper, linear data reconciliation is reformulated by maximizing the information entropy to obtain probability distributions of the data with the minimum incorporation of prior knowledge. Then the reconciled measurements are obtained by maximum likelihood, subject to the process constraints. Two cases are presented, first with only the bounds on the data being specified and second with the variance-covariance matrix of the data additionally being specified. The first case provides a means of performing data reconciliation in the absence of information on the variance-covariance matrix of the data. In the second case, reconciliation using maximum likelihood is formally identical to the conventional least-squares solution. The least-prejudiced probability distribution is a truncated normal distribution, which for reasonably precise data essentially coincides with the multivariate normal distribution. A major difference from conventional reconciliation is that by assuming prior bounds on the measurements, one also should apply those bounds to the reconciled values. An example is used to illustrate the practical implications of the two cases.