so long as the m-by-m matrix exists; this is the case for any m if, for instance, is positive definite. Physically the reason for this property is that since is now a random variable, it is possible to form a meaningful estimate (namely its mean) even with no measurements. Every new measurement simply provides additional information which may modify our original estimate. Another feature of this estimate is that for m < n, there need be no measurement error. Thus, we may have , because as long as is positive definite, the estimate still exists. Lastly, this technique can handle cases where the noise is correlated. Alternative form An alternative form of expression can be obtained by using the matrix identity which can be established by post-multiplying by and pre-multiplying by to obtain and Since can now be written in terms of as , we get a simplified expression for as In this form the above expression can be easily compared with weighed least square and Gauss–Markov estimate. In particular, when , corresponding to infinite variance of the apriori information concerning , the result is identical to the weighed linear least square estimate with as the weight matrix. Moreover, if the components of are uncorrelated and have equal variance such that where is an identity matrix, then is identical to the ordinary least square estimate. Sequential linear MMSE estimation In many real-time applications, observational data is not available in a single batch. Instead the observations are made in a sequence. One possible approach is to use the sequential observations to update an old estimate as additional data becomes available, leading to finer estimates. One crucial difference between batch estimation and sequential estimation is that sequential estimation requires an additional Markov assumption. In the Bayesian framework, such recursive estimation is easily facilitated using Bayes' rule. Given observations, , Bayes' rule gives us the posterior density of as The is called the posterior density, is called the likelihood function, and is the prior density of k-th time step. Note that the prior density for k-th time step is the posterior density of (k-1)-th time step. This structure allows us to formulate a recursive approach to estimation. Here we have assumed the conditional independence of from previous observations given as This is the Markov assumption. The MMSE estimate given the th observation is then the mean of the posterior density . Here, we