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Multiplication of conditional probability with several variables

  1. Aug 6, 2014 #1
    Dear All,

    I am a starter to machine learning and i am currently confused about the following problem:

    what is the result of P(X|Y)P(Y|Z)?
    In my book, it is written to be P(X|Z). But I don't think it is correct since
    P(X|Z)= P(X|Y,Z)P(Y|Z)
    But clearly P(X|Y)=/= P(X|Y,Z)

    Assuming all Events are not independent.

    I have simplified the problem in the above equation. The true equation is
    p(w|x,t,α,β)proportional to p(t|x,w,β)p(w|α) from pattern recognition and machine learning written by christopher m. bishop.

    Any helps and ideas will be very appreciated.
     
  2. jcsd
  3. Aug 6, 2014 #2

    Stephen Tashi

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    Are you saying the above is given as a special condition in the problem?

    Or did you mean [itex] P( \ ( X \cap Y) | Z\ ) = P(X | \ (Y \cap Z)\ )\ P(Y | Z) [/itex] ?
     
  4. Aug 6, 2014 #3

    yes you are correct.
    what I mean is P(x,y|z)=P(x|y,z)P(y|z)
     
  5. Aug 6, 2014 #4

    Stephen Tashi

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    I don't see why that would be correct. Perhaps you need to explain the entire context for it. I don't have a copy of Bishop's book.
     
  6. Aug 7, 2014 #5
    It is in the introduction chapter of the book and is talking about polynomial curve fitting.
    X,T refer to a training set while t refers to the predicted point at position x
    W refers to the set of parameters of M-order polynomial, that
    y(x,w) = w0 + w1*x + w2*x^2 + . . . + wM*x^M

    it claims the following equation for the prediction of t with help of the training set and position x
    p(t|x, X, T) =[itex]\int [/itex] p(t|x,w)p(w|X, T) dw

    that means p(t|x,w)p(w|X, T)= p(t,w|x,X, T) for later maginalization
    But I believe that p(t|x,w)=/= p(t|x,w,X, T)

    If it is not clear enough, i can explain more
     
  7. Aug 7, 2014 #6

    Stephen Tashi

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    To make sense of an expression denoting a probability, we must understand what the "probability space" is. Can you describe the space associated with the notation p(t,x,X,T) ? Is it possible that some of those variables are not random variables, but ordinary variables instead? For example, if I have 3 loaded dice then I might use the notation
    p( X,k)
    to mean "the probability of getting a result of X when I roll the k-th die".. That interpretation doesn't imply that "k" is a random variable. It doesn't implay that there is an experiment where I pick a die at random.
     
  8. Aug 7, 2014 #7
    Let me clarify what you mean: in the expression p(x|m,n), it is not necessary that m and n are random variable. They can be parameters. Whether one is a random variable depend on the setting of the experiment,right?
    IN your case, k can be random variable and p(x,k) means getting a x at random and rolling the k die at random if the experiment is set to be this way.

    I am not sure when it comes to my case.
    In my case, the notation p(t|x, X, T)means
    given the training set X,T and the position x, the probability of finding t. t is obviously random variable. But x,X,T can also be parameters. It is not explicitly written that they are random variables or parameters. The experiment can be predicting t at position x, given a fixed set of X,T. Or the experiment can be predicting t while picking x,X,T at random and now considering P(t|x,X,T). I don't know which experiment the author is doing.
     
  9. Aug 7, 2014 #8

    Stephen Tashi

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    The fact that a p(....) notation can be interpreted in variouis ways, doesn't mean that an equation using it will be correct for each possible interpretation. I suppose an author might use ambiguous notation to assert that a whole family of equations are correct by writing one equation. In your case, I'll guess the author only has one specific interpretation in mind.

    One way to make sense of:

    [itex] p(t|x, X, T) = \int p(t|x,w) p(w,X,T) dw [/itex]

    is to consider [itex] X,T [/itex] to be ordinary variables, not random variables. So within the equation [itex] X,T [/itex] can be treated as if they have some constant value.

    The random variable [itex] t [/itex] is a function only of the random variables [itex] x [/itex] and [itex] w [/itex]
    (i.e [itex] t = w_0 + w_1x + ... w_n x^n [/itex]). So the notation [itex] p(t|x,w) [/itex] means the same thing as [itex] p(t|x,w,X,T) [/itex] because [itex] t [/itex] has no random variation due to [itex] X, T [/itex].

    But by that interpretation, the author could have written [itex] p(w | X,T) [/itex] as [itex] p(w) [/itex]. I supposed he needed to mention [itex] X, T [/itex] somewhere on the right hand side.

    Leaving [itex] X,T [/itex] unmentioned, it isn't controversial that

    [itex] p(t|x) = \int p(t|x,w) p(w) dw [/itex]

    or, mentioning them everywhere, that

    [itex] p(t|x,X,T) = \int p(t|x,w,X,T) p(w| X,T) dw [/itex]
     
  10. Aug 8, 2014 #9
    Thanks so much.I may try to proceed in this direction and see if anything weird occur again.
     
  11. Aug 8, 2014 #10
    I have another question.
    if the above equations are needed to be considered with the following equation.
    p(w|X, T, α, β) ∝ p(T|X,w, β)p(w|α).------(a)
    α, β are fixed.

    The left hand side p(w|X,T) is posterior probability. The right hand side p(w) is the prior probability.
    So X,T are random variables. Right?
    In the book, it mentions that p(w|X,T) in the integral will be given by (a)
     
  12. Aug 8, 2014 #11

    Stephen Tashi

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    It isn't possible to interpret equations without some context. Establishing the context requires a verbal explanation.
    A person who is familiar with the type of problem that Bishop is solving might understand his notation, but I haven't read a statement of what these equations are supposed to accomplish.

    An elementary question that needs a verbl explanation is whether the p(...) notation is supposed to indicate the probability of an event or whether it supposed to denote a probability density function evaluated somewhere. (The value of a a density function evaluated at a point isn't equal to "the proability of" that point.)
     
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