In probability theory, de Finetti's theorem† explains why exchangeable observations are conditionally independent given some latent variable to which an epistemic probability distribution would then be assigned. It is named in honor of Bruno de Finetti.
It states that an exchangeable sequence of Bernoulli random variables is a "mixture" of independent and identically distributed (i.i.d.) Bernoulli random variables – while the individual variables of the exchangeable sequence are not themselves i.i.d., only exchangeable, there is an underlying family of i.i.d. random variables.
Thus, while observations need not be i.i.d. for a sequence to be exchangeable, there are underlying, generally unobservable, quantities which are i.i.d. – exchangeable sequences are (not necessarily i.i.d.) mixtures of i.i.d. sequences.
All of this actually has importance. When we start to assess risks using probabilistic process based on statistical processes, we need to be very careful to understand the underlying mathematics.
There are four approach to saying what we mean when we say “probability”
- Logical – weak implications
- Propensity – physical properties
- Frequency – attributed to sequences of observations
- Subjective – personal opinion
† Finetti’s Theorem is at the heart of estimating random variables. Cost is a random variable, like schedule durations and the technical outcomes from the effort based on cost and schedule. In statistical assessment of cost and schedule, Frequentist (counting) statistics is one approach. The second is Bayesian inference (used in most science). The exchangeability of the random variables is critical to building times series of sampled data from the project to forecast future performance.
The Bigger Problem in Estimating