[R] pmnorm: probabilites don't sum up to 1

peter dalgaard pdalgd at gmail.com
Wed Dec 15 14:05:51 CET 2010

On Dec 15, 2010, at 11:40 , Nicolas Berkowitsch wrote:

> Dear list member,
> I struggle with the problem, why the probabilities of choosing one of three mutually exclusive alternatives don?t sum up to 1!
> Let?s assume we have three alternatives X, Y, and Z. Let?s further assume we know their respective utilities:
> uX, uY, uZ. I?m interested in calculating the probability of choosing X, Y, and Z.
> Since I assume that the alternatives are mutually exclusive, the probabilities p(X), p(Y), and p(Z) have to sum up to one. The utilities of the 3 alternatives can be expressed in 2 utility differences and, hence, the multivariate case reduces to a bivariate normal distribution. If I assume that X, Y, and Z are independent, their corresponding correlations have to be zero and, hence, the variance-covariance-matrices are set to be a diagonal-matrix (i.e., identity-matrix).
> To calculate p(X), p(Y), and p(Z) I was using the following R-code:
> library(mnormt) # can handle multivariate normal distributions
> uX = 2
> uY = 1
> uZ  = .5
> mu = c(uX, uY, uZ)
> LX = matrix(c(1,-1,0,1,0,-1), 2, 3, byrow = TRUE)
> LY = matrix(c(-1,1,0,0,1,-1), 2, 3, byrow = TRUE)
> LZ = matrix(c(-1,0,1,0,-1,1), 2, 3, byrow = TRUE)
> muX = LX %*% mu
> muY = LY %*% mu
> muZ = LZ %*% mu
> Sigma = diag(2)
> mean = c(0,0)
> pX = pmnorm(muX, mean, Sigma)
> pY = pmnorm(muY, mean, Sigma)
> pZ = pmnorm(muZ, mean, Sigma)
> pX + pY + pZ
> I don?t see why the three probabilities don?t sum up to 1?
> I know two ?solutions? to this problem so far. However, neither of them satisfies me:
> 1.	I can set pZ to 1 ? pX ? pY, but doing so, returns a different result for pZ, than calculating pZ directly using pmnorm.
> 2.	I could calculate the relationship of pX to the sum of pX + pY + pZ (? pX/(pX + pY + pZ)   )
> Can anyone explain to me why the probabilities don?t sum up to 1? How should I rewrite the R-code to overcome this problem?
> Thanks a lot for any advice!

I don't think the pX, pY, pZ are probabilities of choosing X, Y, Z. If you think that they are, then you need to explain it more convincingly.

What they are are probabilities of three lower-left quadrants with different origins (muX, muY, muZ). Such quadrants will in general overlap, so there is no reason to expect their probabilities to sum to any particular value. If you were expecting to have a partition of 2d space into three disjoint regions and calculating the probability of each region, then pmnorm is not the right tool. 

Peter Dalgaard
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com

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