Motivation
"We're right 50.75% of the time... but we're 100% right 50.75% of the time. You can make billions that way," said Robert Mercer of Renaissance Technologies.
But what exactly proves that we are 100% right 50.75% of the time? And to what extent can we claim, based on which theory, that being right 50.75% of the time is actually correct?
The model I propose here is the "probability of probability." In other words, it aims to determine "what is the probability that the 50.75% probability is correct?"
The concepts of Delta (Δ) and Gamma (Γ) introduced by the Black-Scholes model successfully compressed information in the financial domain and provided a generalized, inventory-based worldview. This meant that profits and losses arising from fluctuations in held assets could be easily calculated without having to perform base currency calculations each time.
Applying this concept to probability, I have attempted to design a model where what corresponds to Delta (the probability of event A occurring) is termed "first-order probability," and what corresponds to Gamma (the probability that the first-order probability is correct) is termed "second-order probability."
Model
A: Event
Δ(t) ∈ [0,1]: First-order probability at time t (probability that event A occurs)
Γ(t): Second-order probability at time t (probability distribution of Δ being realized)
r(t) ∈ (0, 1]: Range of variation in observable quantitative values at time t (ex. price (magnitude of volatility), weather (3-state range: sunny-cloudy-rainy)) 0<r<1
Δ(t+n): First-order probability at time t+n (probability that event A occurs at time t+n)
α(t) = Δ(t)^(1-r(t)) × 0.5^r(t): Strength of conviction toward "event A occurring"
β(t) = (1-Δ(t))^(1-r(t)) × 0.5^r(t): Strength of conviction toward "event A not occurring"
Assumptions
- r is the only explanatory variable in the system.
- Γ evolves as a function of r.
- Δ evolves as a function of Γ.
- Δ at time t+n is determined by exogenous r.
- Γ(t) is treated as a probability density function over Δ.
- The process satisfies the Markov property (the state at t depends only on information up to t−1).
Formulation Using Beta Distribution Transformation from r(t) to Γ(t):
Γ(t) ~ Beta(α(t), β(t))Transformation from Γ(t) to Δ(t+n):
Δ(t+n) = E[Γ(t)] = α(t)/(α(t) + β(t))Direct formula from r(t) to Δ(t+n):
Δ(t+n) = Δ(t)^(1-r(t)) / [Δ(t)^(1-r(t)) + (1-Δ(t))^(1-r(t))]※ When r(t)=1, α(t)=β(t)=0.5. Beta(0.5, 0.5) is a U-shaped distribution, where (0.5, 0.5) is intended to represent "a state of complete uncertainty"
※ In financial markets, r(t) = σ(t)