Interpreting Probability Adjustments Under Normalization Constraints

Summary

The document asks how to interpret constraints for changing the probabilities of a discrete distribution while retaining its possible outcomes. One condition makes each probability adjustment proportional to the reciprocal of the distance between that outcome and a target expected value, with a scalar parameter controlling the adjustment. A second condition requires the total probability change to sum to zero, preserving normalization.

The answer observes that the two conditions impose a compatibility requirement on the original set of outcomes and the target value. If that requirement holds, the scalar may be chosen freely subject to probability bounds; probabilities both before and after adjustment must remain between zero and one. This can restrict the allowable parameter range. The discussion is algebraic and brief: it does not lay out an optimization algorithm or show a numerical example, and the feasibility conditions depend on the original distribution and the chosen target.

Key ideas

• The adjustment rule makes each probability change depend on its outcome’s distance from the target mean.
• The zero-sum condition on probability changes preserves the total probability mass.
• Together, the stated restrictions require a compatibility condition involving the outcomes and target value.
• Probability bounds before and after adjustment constrain the allowable scalar parameter.
• If the conditions already hold, the setup may leave no substantive optimization choice beyond selecting a feasible parameter.

Tags

# What is the meaning of the following mathematical equations?


# What is the meaning of the following mathematical equations?












Let's say that we have a discrete probability distribution, where

$$ x_i $$ represents each of the possible outcomes (discrete set of possible outcomes), and

$$ L $$ represents the expected value we want to achieve, by manipulating the original probability distribution (keeping the same values of x_i, but adjusting the corresponding probabilities.

To do so, I am asked to use an optimization method, while implementing the two following restrictions,

$$ \Delta P(x_i) = \alpha /(x_i - L) $$

$$ \Delta \sum_{i=-n}^{n} P(x_i) = 0 $$

How should I interpret the above restrictions, so I can implement them computationally?

## Answer by Pontus Hultkrantz (score 1)

https://quant.stackexchange.com/a/73300

First condition says how much you change the probability mass as a function of alpha. Second one that probability must sum up to one. The two conditions imply that $\sum_i (x_i-L)^{-1}=0$ for thr original problem, and that the mean $L \neq x_j$ for all $j$. If this is originally fulfilled, you are free to chose any real $\alpha$, so not much of an optimization. However, you should also add that each probability is in $[0,1]$ before and after, which will bound your possible alpha.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.