How to calculate the mean of two posterior probabilities using Bayes' theorem?

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I'm doing atmospheric retrievals using non-linear optimal estimation, and I would like to calculate the associated uncertainty on the mean of two atmospheric profiles each starting from a different prior solution, but I'm not entirely sure how to do this in the most mathematically rigorous way possible.



My known variables are:



- $P(x_1|y)$, which is given by the covariance matrix $mathbfhatS_1$
- $P(x_2|y)$, which is given by the covariance matrix $mathbfhatS_2$
- $P(x_1)$, which is given by the prior covariance matrix $mathbfhatS_a1$
- $P(x_2)$, which is given by the prior covariance matrix $mathbfhatS_a2$


and I want to calculate:
$P(fracx_1 + x_22 | y)$




probability bayes-theorem inverse-problems




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    I'm doing atmospheric retrievals using non-linear optimal estimation, and I would like to calculate the associated uncertainty on the mean of two atmospheric profiles each starting from a different prior solution, but I'm not entirely sure how to do this in the most mathematically rigorous way possible.



    My known variables are:



    - $P(x_1|y)$, which is given by the covariance matrix $mathbfhatS_1$
    - $P(x_2|y)$, which is given by the covariance matrix $mathbfhatS_2$
    - $P(x_1)$, which is given by the prior covariance matrix $mathbfhatS_a1$
    - $P(x_2)$, which is given by the prior covariance matrix $mathbfhatS_a2$


    and I want to calculate:
    $P(fracx_1 + x_22 | y)$




    probability bayes-theorem inverse-problems




    share|cite|improve this question
























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      I'm doing atmospheric retrievals using non-linear optimal estimation, and I would like to calculate the associated uncertainty on the mean of two atmospheric profiles each starting from a different prior solution, but I'm not entirely sure how to do this in the most mathematically rigorous way possible.



      My known variables are:



      - $P(x_1|y)$, which is given by the covariance matrix $mathbfhatS_1$
      - $P(x_2|y)$, which is given by the covariance matrix $mathbfhatS_2$
      - $P(x_1)$, which is given by the prior covariance matrix $mathbfhatS_a1$
      - $P(x_2)$, which is given by the prior covariance matrix $mathbfhatS_a2$


      and I want to calculate:
      $P(fracx_1 + x_22 | y)$




      probability bayes-theorem inverse-problems




      share|cite|improve this question














      I'm doing atmospheric retrievals using non-linear optimal estimation, and I would like to calculate the associated uncertainty on the mean of two atmospheric profiles each starting from a different prior solution, but I'm not entirely sure how to do this in the most mathematically rigorous way possible.



      My known variables are:



      - $P(x_1|y)$, which is given by the covariance matrix $mathbfhatS_1$
      - $P(x_2|y)$, which is given by the covariance matrix $mathbfhatS_2$
      - $P(x_1)$, which is given by the prior covariance matrix $mathbfhatS_a1$
      - $P(x_2)$, which is given by the prior covariance matrix $mathbfhatS_a2$


      and I want to calculate:
      $P(fracx_1 + x_22 | y)$





      probability bayes-theorem inverse-problems





      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 24 at 13:41









      Tommi Brander

      936822




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      asked Aug 22 at 11:03









      user68150

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