Error estimation with boundary condition

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Given a multiset of elements M.



I would like to compute the relative likelihood of the elements within M.



Let c be the count of an element, meaning how often it is contained in M.



Then the relative likelihood p is p = c/|M|.



However I only have M' which is a subset of M. Every apearence of an element in M is missing in M' with a probability of 0 < x < 1. By this I mean if an element is contained in M 3 times each of the three times can be missing individually with the chance x.



Calculating the relative likelihood for this M' now propagates this error.



So the size of M' is (1-x)|M| and the count c of an element i should be (1-xi)c. The error of the count of an element will obviously not be x for all of them.



Because I am calculating relative likelihoods I have the boundary condition that the sum of all p over all elements needs to be 1. Or equivalently that the sum of all counts (with error) needs to be equal to all elements in the multiset (1-x)|M|.



How would one estimate the error for the relative likelihoods in this scenario?




probability error-propagation




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    Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 30 at 9:34














up vote
0
down vote

favorite












Given a multiset of elements M.



I would like to compute the relative likelihood of the elements within M.



Let c be the count of an element, meaning how often it is contained in M.



Then the relative likelihood p is p = c/|M|.



However I only have M' which is a subset of M. Every apearence of an element in M is missing in M' with a probability of 0 < x < 1. By this I mean if an element is contained in M 3 times each of the three times can be missing individually with the chance x.



Calculating the relative likelihood for this M' now propagates this error.



So the size of M' is (1-x)|M| and the count c of an element i should be (1-xi)c. The error of the count of an element will obviously not be x for all of them.



Because I am calculating relative likelihoods I have the boundary condition that the sum of all p over all elements needs to be 1. Or equivalently that the sum of all counts (with error) needs to be equal to all elements in the multiset (1-x)|M|.



How would one estimate the error for the relative likelihoods in this scenario?




probability error-propagation




share|cite|improve this question















  • 1




    Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 30 at 9:34












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Given a multiset of elements M.



I would like to compute the relative likelihood of the elements within M.



Let c be the count of an element, meaning how often it is contained in M.



Then the relative likelihood p is p = c/|M|.



However I only have M' which is a subset of M. Every apearence of an element in M is missing in M' with a probability of 0 < x < 1. By this I mean if an element is contained in M 3 times each of the three times can be missing individually with the chance x.



Calculating the relative likelihood for this M' now propagates this error.



So the size of M' is (1-x)|M| and the count c of an element i should be (1-xi)c. The error of the count of an element will obviously not be x for all of them.



Because I am calculating relative likelihoods I have the boundary condition that the sum of all p over all elements needs to be 1. Or equivalently that the sum of all counts (with error) needs to be equal to all elements in the multiset (1-x)|M|.



How would one estimate the error for the relative likelihoods in this scenario?




probability error-propagation




share|cite|improve this question











Given a multiset of elements M.



I would like to compute the relative likelihood of the elements within M.



Let c be the count of an element, meaning how often it is contained in M.



Then the relative likelihood p is p = c/|M|.



However I only have M' which is a subset of M. Every apearence of an element in M is missing in M' with a probability of 0 < x < 1. By this I mean if an element is contained in M 3 times each of the three times can be missing individually with the chance x.



Calculating the relative likelihood for this M' now propagates this error.



So the size of M' is (1-x)|M| and the count c of an element i should be (1-xi)c. The error of the count of an element will obviously not be x for all of them.



Because I am calculating relative likelihoods I have the boundary condition that the sum of all p over all elements needs to be 1. Or equivalently that the sum of all counts (with error) needs to be equal to all elements in the multiset (1-x)|M|.



How would one estimate the error for the relative likelihoods in this scenario?





probability error-propagation





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asked Jul 30 at 9:20









Leandro

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  • 1




    Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 30 at 9:34












  • 1




    Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 30 at 9:34







1




1




Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 30 at 9:34




Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 30 at 9:34















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