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I am looking for a statistical method that closely relates to the idea of potential energy.

Here is a quick google definition for potential energy

"...the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors."

The classic example is the gravitational potential energy, for example: the higher you lift a brick - the higher the ability to do work, or apply a force on another object.

Now here is a similar example in statistics. Let's say that we are studying lung cancer and we discriminate on how many cigarettes people used to smoke. If one smokes two packs a day - it is four times more likely to get lung cancer. If one smokes three packs a day - than it is nine times more likely; hence, we can model the "statistical potential" with the function y = x^2 where x - is the number of smokes.

Note that I am not looking for a probability... it must be a positive number from zero to infinity.

The closest thing I have found that resembles my idea is Relative Risk but typically is a binary choice - the group either smokes or it doesn't.

Edit

What is the continuous version or Relative Risk or Odds Ratio?

Edit 2 Here is my take if RR and OR was not a thing

Let A be the event one smoke x-number of cigarettes Let B the event one gets lung diseases Let x be the number of cigs one smokes in a fixed time period

By bayes rule we have

$$ P(A|B) = \frac{P(B|A)P(B)}{ \intop\nolimits_{a}^{b} P(B|A)P(A)dA } $$

But we need some kind of partial expectation where we can see the difference of smoking 0 to x-many cigarettes. Then

$$ V(x) = \intop\nolimits_{0}^{x} x*P(A|B) dx $$

Where $V(x)$ we can call the statistical potential of smoking x-cigarettes

Of course we can pick proper distributions to model the events.

Now - we can see the magnitude of the expectation for $x_1 < x_2 < max_x$

This really sums up my idea of the statistical potential.

Is this acceptable?

Perhaps this should be all the way around - the expectation should be on the consequence but it makes the point

Edit 3

I think I have convinced myself that RR is the way to go about. Here is the deal

Let $RR(1)$ = P(disease|smoke 1 cigs) / P(disease|smoke 0 cigs)

Let $RR(2)$ = P(disease|smoke 2 cigs) / P(disease|smoke 0 cigs) and so on

Then $V(x) = RR(x)$ = P(disease|smoke x cigs) / P(disease|smoke 0 cigs)

One can fit a function between the pairs such as in my example $y=f(x)=V(x)$

I want credit for coining the term "statistical potential energy" :)

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closed as unclear what you're asking by Michael Chernick, user158565, mkt, kjetil b halvorsen, Peter Flom Feb 7 at 11:14

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ What's wrong with odds ratio or relative risk? $\endgroup$ – Peter Flom Feb 7 at 11:13
  • $\begingroup$ @PeterFlom Nothing wrong besides the fact that they are not continuous - I am asking for confirmation that RR fits my idea of potential energy. $\endgroup$ – Edv Beq Feb 7 at 12:06
  • $\begingroup$ They are continuous. The OR and the RR can take any value from 0 to infinity. $\endgroup$ – Peter Flom Feb 7 at 12:09
  • $\begingroup$ @PeterFlom okay thank you - which one do you think is more appropriate? $\endgroup$ – Edv Beq Feb 7 at 12:15
  • $\begingroup$ I don't know. They are usually similar. $\endgroup$ – Peter Flom Feb 7 at 21:35
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If you have a coordinate $x$ that separates the states of your system, and you can assign a probability $P(x)$ to finding your system at coordinate value $x$, then you can define a kind of energy as follows. Define $E(x)$ such that $P(x) \sim \exp(-\beta E(x))$ or $E(x) = -\log(P(x)/\beta + E_0$, where $\beta$ is a constant that can be ignored for now. That gets you an energy-like thing.

But how would you use it? Would you try to define a force $F=-dE(x)/dx$? Would you like to say that the average energy $<E>$ is a constant of the system given the constancy of several intensive features, like $\beta$? Would you try to define dynamics in $x$, or kinetics? or diffusion? I'm genuinely curious where you are going with this . . .

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  • $\begingroup$ Lately, I have been watching a lot of videos of the Schrodinger equation and I thought there should be such a concept in statistics. $\endgroup$ – Edv Beq Feb 7 at 3:28
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    $\begingroup$ Never forget, quantum mechanics works because it does. A fair amount of it was people like Schroedinger saying "what qualities would a wave function likely have if the ideas of quantum mechanics is right", and then reasoning from there. One of the hardest things to teach in QM is there "is no reason" it works - but if you do the math as described, it works. $\endgroup$ – eSurfsnake Feb 7 at 3:42
  • $\begingroup$ I'd just like to warn you that phenomenology is seductive, but tricky. You can use physical analogies to help your thinking, sure, but sooner or later you will violate an assumption in either stats or physics, and the formerly useful analogy will break down. When that happens, don't be too clingy! Let it go . . . ;) $\endgroup$ – Peter Leopold Feb 7 at 14:12
  • $\begingroup$ @PeterLeopold I like your answer - but I am thinking it would have to be some type of conditional expectation between "action" event i.e.: smoking cigarettes, and the "consequence" event i.e.: getting the lung disease. Actually partial conditional expectation. Bayes rule comes to mind too. $\endgroup$ – Edv Beq Feb 8 at 0:55

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