Updating weight based on fisher scoring algorithm [wikipedia]

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My understanding on weight update for fisher is based on wikipedia description. The fisher information is said as the expectation of hessian. Can someone explain how is the expection for each entry of hessian to be found out? I have dropped that idea as I have found the fisher information can also be calculated using covariance of score function(jacobian).

So my question is

The covariance matrix will be a (pxp) matrix where p is the number of attributes. So for β update, I will have to take the inverse of covariance matrix, multiply it with mxp matrix(score fn) and then add it to my β(old)?

Can someone please tell whether this understanding right or not??