Here and below, formula_6 means proportional; a pdf is always scaled so that its integral over the whole space is one. This formula_7, called the "prior", was evolved in time by running the model and now is to be updated to account for new data. It is natural to assume that the error distribution of the data is known; data have to come with an error estimate, otherwise they are meaningless. Here, the data formula_8 is assumed to have Gaussian pdf with covariance formula_9 and mean formula_10, where formula_11 is the so-called observation matrix. The covariance matrix formula_9 describes the estimate of the error of the data; if the random errors in the entries of the data vector formula_8 are independent, formula_9 is diagonal and its diagonal entries are the squares of the standard deviation (“error size”) of the error of the corresponding entries of the data vector formula_8. The value formula_10 is what the value of the data would be for the state formula_1 in the absence of data errors. Then the probability density formula_18 of the data formula_8 conditional of the system state formula_1, called the data likelihood, is