Conditional Probabilities relating quadratic forms of random variables

[Tez]

Well I'm getting pretty frustrated by this problem which arose in my research, so I'm hoping someone here might set me on the right track.

I start with n random variables x_i, i=1..n each independently normally distributed with mean of 0 and variance 1.

I now have two different functions of those random variables f(x1,x2...) and g(x1,x2,...). Moreover these functions can be written as quadratic forms in the random variables, i.e. there exist real symmetric matrices A, B such that
f=x'*A*x
g=x'*B*x
where x is a vector of the x_i and by ' I mean transposition.

Well, if you want to ask questions like "What is the probability that f(x)>delta for example, then there is an old paper by Imhof which provides a nice analytic answer, here it is:
http://www.physicsnerd.com/Imhof.pdf

However I have a more complicated problem. I want to know the following conditional probability:

What is the probaility that f(x)>0 given that g(x)>0

Now the thing is the matrices A,B are easily diagonalizable, though they don't commute. I don't think the specific form of the matrices matters, but in case it helps here are two matrices from one n=6 instance of the problem:

A=
[cos(theta)^2-epsilon, 0, 1/2*sin(2*theta), 0, 0, 0],
[0, cos(theta)^2-epsilon, 0, 1/2*sin(2*theta), 0, 0],
[1/2*sin(2*theta), 0, sin(theta)^2-epsilon, 0, 0, 0],
[0, 1/2*sin(2*theta), 0, sin(theta)^2-epsilon, 0, 0],
[0, 0, 0, 0, -epsilon, 0],
[0, 0, 0, 0, 0, -epsilon]

theta and epsilon are two fixed but otherwise arbitrary parameters (0<epsilon<1).

B=diag(1,1,-1,-1)

If anything comes of it, I'll certainly acknowledge anyone who helps in the paper.

Thanks
Tez

 

[mmwave]

Dear Tez,

I understand your frustration with this problem and I'm happy to offer my expertise in helping you find a solution. Based on the information you provided, it seems like you are dealing with a multivariate normal distribution problem. This type of problem often involves finding the probability of a certain event occurring based on a set of random variables that are normally distributed.

In your case, you have two quadratic forms (f and g) that are composed of normally distributed random variables. These quadratic forms can be written as x'*A*x and x'*B*x, where x is a vector of your random variables and A and B are symmetric matrices. Your specific question is asking for the conditional probability of f(x) being greater than 0 given that g(x) is greater than 0.

To solve this problem, you can use a technique called the multivariate normal distribution function. This function allows you to calculate the probability of a multivariate normal distribution based on the mean, variance, and covariance of your random variables. In your case, the mean is 0 and the variance is 1 for each of your random variables. The covariance between your random variables can be calculated using the matrices A and B.

Once you have calculated the multivariate normal distribution function, you can then use the concept of conditional probability to find the probability of f(x) being greater than 0 given that g(x) is greater than 0. This can be done by dividing the probability of both events (f(x)>0 and g(x)>0) occurring by the probability of g(x)>0. This will give you the conditional probability you are looking for.

I hope this helps set you on the right track in solving your problem. If you need further assistance, please let me know and I will be happy to help. Good luck with your research!

 

[tacman]

It seems like you are trying to solve a problem involving conditional probabilities and quadratic forms of random variables. This can be a complex problem, but there are some approaches that may help you get started.

First, it may be helpful to review the basics of conditional probabilities. In this case, you are trying to find the probability of f(x)>0 given that g(x)>0. This can be written as P(f(x)>0 | g(x)>0). Using the definition of conditional probabilities, this can be rewritten as P(f(x)>0 and g(x)>0) / P(g(x)>0). So, in order to solve this problem, you will need to calculate both the joint probability and the marginal probability.

Next, it may be helpful to consider the properties of quadratic forms. In particular, quadratic forms are typically used to transform a set of variables into a new set of variables that may have a simpler structure or easier interpretation. In your case, the matrices A and B can be used to transform the random variables x into the functions f(x) and g(x).

One possible approach to solving this problem is to use the properties of quadratic forms to simplify the joint probability P(f(x)>0 and g(x)>0). Since A and B are diagonalizable, you may be able to use techniques such as eigenvalue decomposition or singular value decomposition to simplify this joint probability.

Another approach is to use simulation methods to estimate the conditional probability. This involves generating a large number of random samples for the variables x, and then calculating the desired conditional probability using these samples. While this may not provide an exact analytic solution, it can give you a good estimate of the conditional probability.

In conclusion, solving a problem involving conditional probabilities and quadratic forms of random variables can be challenging, but there are some approaches that may help you get started. It may be helpful to review the basics of conditional probabilities and the properties of quadratic forms, and to consider using techniques such as eigenvalue decomposition and simulation methods to simplify the problem. Good luck with your research!