Measuring The RATE of Probability Change

In summary: The mean number is the sum of the probabilities of all possible ways the state can be occupied, times the number of particles which can occupy the state, times the probability of those particles being in the state. In summary, the conversation discusses the measurement of the rate of change of a stochastic process's probability/probability density function. The classic definition of a derivative does not work for stochastic processes, and Ito's calculus and Ito's Lemma are used instead. The discussion also mentions the Stratonovich Integral and the use of Probability Space and the Radon-Nikodym derivative. The conversation suggests using the maximum likelihood method or techniques of Hidden Markov Models or the EM algorithm to estimate the parameters and calculate the rate of change of
  • #1
LarryS
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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.
 
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  • #2
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 defintion.

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.
 
  • #3
referframe said:
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:


[tex]G_{1}=\frac{d\mu_{0}}{dt}[/tex] and [tex]G_{2}=\frac{d\mu_{1}}{dt}[/tex]

Where the [tex]\mu_{0}[/tex] is the population mean and [tex]\mu_{1}[/tex] is the population variance.

then find [tex]F(G_{1},G_{2})[/tex] where F(Gi) is the rate of change of the PDF (F'(x)).
 
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  • #4
referframe said:
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.
 
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  • #5
referframe said:
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
 
  • #6
bpet said:
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.
 
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1. What is the purpose of measuring the rate of probability change?

The purpose of measuring the rate of probability change is to understand how quickly or slowly a certain event or outcome is likely to occur. It helps us make informed decisions and predictions based on the likelihood of a particular event happening within a given timeframe.

2. How is the rate of probability change calculated?

The rate of probability change is calculated by dividing the change in probability by the time interval over which the change occurred. This is usually represented as a percentage or a decimal value.

3. Can the rate of probability change be negative?

Yes, the rate of probability change can be negative. This indicates a decrease in the likelihood of an event occurring over time. For example, if the probability of winning a game decreases from 60% to 50% over a certain period, the rate of probability change would be -10%.

4. What factors can affect the rate of probability change?

The rate of probability change can be affected by a variety of factors, such as external events, changes in circumstances, or new information. It can also be influenced by individual biases and subjective interpretations of data.

5. How can measuring the rate of probability change be useful in decision making?

Measuring the rate of probability change can be useful in decision making by providing a more accurate understanding of the likelihood of a particular event occurring. This can help individuals and organizations make informed decisions, take appropriate actions, and plan for the future based on the level of risk involved.

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