Exchangeability and Dependence Mixtures

Worked Example

Consider a binary sequence of length 3 under a latent probability distribution. For a "Mild" latent law where P=0.25 and P=0.75 are equally likely (weight 0.5 each), the sequence probability for "001" is computed by mixing the IID probabilities under each latent state.

Let's trace the calculation for the sequence "001":

1. For P=0.25, the IID probability is (0.75)²(0.25) = 0.140625.
2. For P=0.75, the IID probability is (0.25)²(0.75) = 0.046875.
3. The expected mixture probability is 0.5 * 0.140625 + 0.5 * 0.046875 = 0.09375.

Experiment

In this Interactive Module, you can select different latent laws and observe how the sequence and count probabilities change.

To see how these concepts behave in practice, try the following steps:

1. Select the "Mild" latent law and observe the IID comparison values.
2. Select the "Finite one success" latent law to observe a distribution that is finite exchangeable over length 3, and note the negative covariance.

Limitations

The visualizations and calculations in this module are strictly limited to binary sequences of length 3. Real-world applications of exchangeability often involve continuous variables or much longer sequences where exact computation is replaced by Markov Chain Monte Carlo (MCMC) methods or approximations. Additionally, the specific finite exchangeability case presented is an extreme edge case designed for theoretical illustration.

References

For further reading on the boundary of finite and infinite exchangeable probabilities, refer to the theoretical framework in classical probability literature:

• Finite Exchangeable Probability (Stanford Technical Reports)
• Aldous, D. J. (1985). Exchangeability and related topics. École d'Été de Probabilités de Saint-Flour XIII.