Probability via Expectation Measures.

Area of Science:

• Probability theory
• Measure theory
• Statistics

Background:

• Traditional probability theory, based on Kolmogorov's work, uses measures with total mass 1 to model single outcomes.
• Kolmogorov's model assumes an experiment's outcome belongs to exactly one of several disjoint sets.

Purpose of the Study:

• To present an alternative basic model for probability experiments.
• To shift the theoretical focus from probabilities to expected values.
• To demonstrate the utility of this new framework in various scientific domains.

Main Methods:

• Developed a new framework where experiments yield multisets (counts of observations in each set).
• Ensured consistency with Kolmogorov's measure-theoretic probability theory.
• Derived new theorems to support the theoretical framework.

Main Results:

• The new model focuses on expected values, offering a different perspective than traditional probability measures.
• Demonstrated applications and benefits in goodness-of-fit testing, Bayesian statistics, and quantum theory.
• New theorems were established to address challenges arising from the shift in focus.

Conclusions:

• The multiset-based framework provides a consistent alternative to traditional probability theory.
• Shifting focus to expected values offers new insights and enhanced performance in applied fields.
• This approach broadens the foundational understanding and application of probability theory.