How Does the Form of P(x) Affect Probabilistic Methods in Calculating Moments?

[eljose]

I mean if you define the n-th momentum of a probability distribution:

$$<x^n > = \int_{-\infty}^{\infty} dx P(x) x^n$$ (1)

then my question is about the function P(x) (probability distribution) when:

$$P(x)=e^{-af(x) }$$ where a is a constant (real or complex)

-if f(x) is f(x)=x^2 then you have a "Gaussian" for a>0 and you can obtain every moment using (1)

- if f(x) is any function and a>0 ,a-->oo (big) you can use "Saddle-point approximation".

-the question is how can you handle if P(x) is of the form:

$$P(x)=x^2 + \alpha g(x)$$ where "alpha" is an small coupling constant so you could expand $$exp(\alpha g(x) )$$ and take only a few terms in that case i think you can take:

$$\int_{-\infty}^{\infty} dx P(x) = \sum_ n a(n) <x^n >$$

 

[mmwave]

I would like to provide some insight into the question posed by the forum post. Firstly, the function P(x) in the given equation (1) represents the probability distribution of a random variable x. This means that for a given value of x, P(x) gives the probability of that value occurring. In the case of a Gaussian distribution, f(x) is simply x^2, which is why the n-th momentum can be obtained using the given equation.

Now, if P(x) is of the form P(x)=x^2 + \alpha g(x), where \alpha is a small coupling constant and g(x) is some arbitrary function, we can still use the same equation (1) to calculate the n-th momentum. However, in this case, the integral becomes more complicated and cannot be solved analytically. This is where the concept of saddle-point approximation comes into play.

Saddle-point approximation is a method used to approximate a complex integral by replacing it with a simpler integral that can be solved analytically. In this case, we can use the saddle-point approximation to simplify the integral and obtain the moments of the distribution.

The idea behind saddle-point approximation is to find the point in the complex plane where the integrand reaches its maximum value. This point is known as the saddle point, and is denoted by x_s. We can then expand the integrand around this saddle point and keep only a few terms to obtain an approximation of the integral.

Using this method, we can write the integral as:

\int_{-\infty}^{\infty} dx P(x) \approx P(x_s) \int_{-\infty}^{\infty} dx e^{-\frac{1}{2} (x-x_s)^2 P''(x_s)},

where P''(x_s) is the second derivative of P(x) evaluated at the saddle point x_s. This approximation is known as the Gaussian approximation, and it becomes more accurate as \alpha becomes larger.

In summary, as a scientist, I would approach the problem of handling P(x) of the form P(x)=x^2 + \alpha g(x) by using the saddle-point approximation method to simplify the integral and obtain the moments of the distribution. This method is useful in cases where the integral cannot be solved analytically, and it becomes more accurate as the coupling constant \alpha increases.

 

[tacman]

It seems that you have identified a potential issue with using probabilistic methods in certain situations. While the definition of the n-th momentum of a probability distribution may seem straightforward, the function P(x) can present challenges in certain cases. Specifically, when P(x) is of the form x^2 + alpha g(x), where alpha is a small coupling constant, using the Saddle-point approximation may not be sufficient. In these cases, it may be necessary to expand the exponential term and take only a few terms, which can lead to a different result.

One possible solution to this problem could be to use a different method, such as the Monte Carlo method, which involves generating random samples from the distribution and using these samples to approximate the desired moment. This method can be more accurate in cases where the P(x) function is not easily tractable.

Another approach could be to use numerical integration techniques, such as Gaussian quadrature, to approximate the integral. This can also provide more accurate results in cases where the P(x) function is complex.

Overall, it is important to carefully consider the nature of the P(x) function when using probabilistic methods and to explore different techniques to ensure accurate results.