Measuring The RATE of Probability Change

[LarryS]

Suppose that we have a stochastic process and we suspect that the probability/probability density function changes with time, changes slowly in relation to the sampling rate.

Two examples might be:

1. A QM experiment in which the source of identically prepared particles is not quite a stationary process.
2. Repeated rolling of a hypothetical/magical 6-faced die for which the probability for the faces is not uniform and changes slowly with time.

From a practical standpoint, how would we measure how fast the overall probability function is changing?

(I do not have a strong background in statistics).

As always, thanks in advance.

 

[Oxymoron]

Just as in Classical-calculus you define the derivative (the rate of change) as the inverse of integration (area under a curve) via the Fundamental Theorem of Calculus - or through the use of limits. However, this definition is not suitable for Stochastic Calculus because (now I am doing some hand-waving here) when you take a Taylor Expansion of a Random Variable the Delta-x's tend to the variance of the distribution and NOT to zero as we would see in classical calculus. Therefore the classic definition of a derivative does not work for stochastic processes and we need a new definition.

This is the basis behind Ito's calculus, and, in particular, Ito's Lemma which allows you to calculate the the derivative of a stochastic process.

I am not too sure about 1. but I would suggest using the Stratonovich Integral. I have never used it but I've heard it is useful for physics problems.

For 2. however, rolling a magical die such as this would fall right into the scope of Ito Calculus. I believe it is something you can model this with a Probability Space and then use the Radon-Nikodym derivative.

 

[SW VandeCarr]

From a practical standpoint, how would we measure how fast the overall probability function is changing?

(I do not have a strong background in statistics).

As always, thanks in advance.

It seems you're really concerned with the rate of change of the sample parameters; that is, the mean and variance. These are independent variables in your sampling distribution assuming the population has a Gaussian (normal) distribution. From your estimates of the these population parameters, define the functions:


$$G_{1}=\frac{d\mu_{0}}{dt}$$ and $$G_{2}=\frac{d\mu_{1}}{dt}$$

Where the $$\mu_{0}$$ is the population mean and $$\mu_{1}$$ is the population variance.

then find $$F(G_{1},G_{2})$$ where F(Gi) is the rate of change of the PDF (F'(x)).

 

[SW VandeCarr]

Suppose that we have a stochastic process and we suspect that the probability/probability density function changes with time, changes slowly in relation to the sampling rate.

Note we are not necessarily talking about a non stationary stochastic process here. If the process were Markovian (probabilistic transitions of population states) we could use a transition probability matrix. The functional notation would then simply describe some generalization or expectation of the population's evolution. However, without that assumption, the alternative is that the evolution of the population parameters is (macroscopically) deterministic.

 

[bpet]

From a practical standpoint, how would we measure how fast the overall probability function is changing?

If the distribution evolves according to some deterministic rule, e.g. P[X(k)=6]=a*k+b for the dice example, then the parameters (a,b) can be estimated from observed dice rolls by the maximum likelihood method, and the rate of change of probability follows directly.

If the distribution evolves according to a non-deterministic rule (e.g. P[X(k)=6]=P[X(k-1)=6]+c*Z(k) where Z is random) then it should still be possible but may require techniques of Hidden Markov Models or the EM algorithm.

HTH

 

[SW VandeCarr]

If the distribution evolves according to some deterministic rule, e.g. P[X(k)=6]=a*k+b for the dice example, then the parameters (a,b) can be estimated from observed dice rolls by the maximum likelihood method, and the rate of change of probability follows directly.

If the distribution evolves according to a non-deterministic rule (e.g. P[X(k)=6]=P[X(k-1)=6]+c*Z(k) where Z is random) then it should still be possible but may require techniques of Hidden Markov Models or the EM algorithm.

HTH

Based on the examples the OP gave, I was thinking in terms of physical processes such as in thermodynamics. Temperature (absolute) is the average kinetic energy of molecules (or simply 'particles') in a sample. This kinetic energy follows a distribution at the particle level. There are various statistics which apply depending on the physical characteristics of the substance. However, at the macro level the change in temperature with respect to change in energy follows deterministic laws in ideal gases and in pure substances where the specific heat is known for various phase states. In the OP's case it seems a relatively small number of "prepared" particles are involved, so the evolution of the PDF may be important. In general, Fermi-Dirac statistics are calculated in terms of the mean number of particles in a given state.