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Thread: How do I derive Conditional distribution from the marginal distributions?

  1. #1

    How do I derive Conditional distribution from the marginal distributions?

    I am not very sure if I am able to pose the question properly, but I am having a lot of problem trying to justify a joint distribution derived from the marginals.

    I am given that:

    (yt+1 | xt+1, θ) ~ N (xt+1, σ2)
    (xt+1 | xt, θ) ~ N (xt+1, σ2)
    (yt+1 | xt, θ) ~ N (xt, σ2 + τ2)

    (Basically θ represents the parameters sigma and tau)

    Then how can we derive (xt+1 | yt+1, xt, θ)?

    The answer is:
    (xt+1 | yt+1, xt, θ) ~ N (µ, w2),
    µ = w2-2yt+1 + τ-2xt) and w-2 = σ-2+ τ-2

    but I don’t know how to calculate it. Any help will be much appreciated.

  2. #2
    Elite Member
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    Quote Originally Posted by shehryaramin View Post
    I am not very sure if I am able to pose the question properly, but I am having a lot of problem trying to justify a joint distribution derived from the marginals.

    I am given that:

    (yt+1 | xt+1, θ) ~ N (xt+1, σ2)
    (xt+1 | xt, θ) ~ N (xt+1, σ2)
    (yt+1 | xt, θ) ~ N (xt, σ2 + τ2)

    (Basically θ represents the parameters sigma and tau)

    Then how can we derive (xt+1 | yt+1, xt, θ)?

    The answer is:
    (xt+1 | yt+1, xt, θ) ~ N (µ, w2),
    µ = w2-2yt+1 + τ-2xt) and w-2 = σ-2+ τ-2

    but I don’t know how to calculate it. Any help will be much appreciated.
    What are your thoughts?

    Please share your work with us ...even if you know it is wrong.

    If you are stuck at the beginning tell us and we'll start with the definitions.

    You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

    http://www.freemathhelp.com/forum/announcement.php?f=33
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
    Quote Originally Posted by Subhotosh Khan View Post
    What are your thoughts?

    Please share your work with us ...even if you know it is wrong.

    If you are stuck at the beginning tell us and we'll start with the definitions.

    You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

    http://www.freemathhelp.com/forum/announcement.php?f=33


    I think to get at the distribution I use the conditional probability rule to first get x(t+1) conditional on y(t+1) and y(t+1) conditional on x(t) and theta.

    But I am not sure if that is the right procedure, And still if I get that, I have problems deriving x(t+1) conditional on y(t+1).

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