Entropy program will change the initial predictive distribution 'p' to a new set 'p_' that satisfies specified moment conditions but changes other propoerties of the new distribution the least by minimizing the relative entropy between the two distributions. Theoretical note: Relative Entropy (Kullback-Leibler information criterion KLIC) is an asymmetric measure.
EntropyProg(p, A = NULL, b = NULL, Aeq, beq, verbose = FALSE)a list with
p_:revised probabilities based on entropy pooling
optimizationPerformance:a list with status of optimization, value, number of iterations, and sum of probabilities
a vector of initial probabilities based on prior (reference model, empirical distribution, etc.). Sum of 'p' must be 1
matrix consisting of inequality constraints (paired with argument 'b'). Denoted as 'F' in the Meucci paper
vector consisting of inequality constraints (paired with matrix A). Denoted as 'f' in the Meucci paper
matrix consisting of equality constraints (paired with argument 'beq'). Denoted as 'H' in the Meucci paper. (denoted as 'H' in the "Meucci - Flexible Views Theory & Practice" paper formlua 86 on page 22)
vector corresponding to the matrix of equality constraints (paired with argument 'Aeq'). Denoted as 'h' in the Meucci paper
If TRUE, prints out additional information. Default FALSE.
' $$ \tilde{p} \equiv argmin_{Fx \leq f, Hx \equiv h} \big\{ \sum_1^J x_{j} \big(ln \big( x_{j} \big) - ln \big( p_{j} \big) \big) \big\} \\ \ell \big(x, \lambda, \nu \big) \equiv x' \big(ln \big(x\big) - ln \big(p\big) \big) + \lambda' \big(Fx - f\big) + \nu' \big(Hx - h\big)$$
Ram Ahluwalia ram@wingedfootcapital.com
We retrieve a new set of probabilities for the joint-scenarios using the Entropy pooling method Of the many choices of 'p' that satisfy the views, we choose 'p' that minimize the entropy or distance of the new probability distribution to the prior joint-scenario probabilities.
We use Kullback-Leibler divergence or relative entropy dist(p,q): Sum across all scenarios [ p-t * ln( p-t / q-t ) ] Therefore we define solution as p* = argmin (choice of p ) [ sum across all scenarios: p-t * ln( p-t / q-t) ], such that 'p' satisfies views. The views modify the prior in a cohrent manner (minimizing distortion) We forumulate the stress tests of the baseline scenarios as linear constraints on yet-to-be defined probabilities Note that the numerical optimization acts on a very limited number of variables equal to the number of views. It does not act directly on the very large number of variables of interest, namely the probabilities of the Monte Carlo scenarios. This feature guarantees the numerical feasability of entropy optimization.
Note that new probabilities are generated in much the same way that the state-price density modifies objective probabilities of pay-offs to risk-neutral probabilities in contingent-claims asset pricing
Compute posterior (=change of measure) with Entropy Pooling, as described in
A. Meucci - "Fully Flexible Views: Theory and Practice". See page 22 for illustration of numerical implementation Symmys site containing original MATLAB source code https://www.arpm.co/ NLOPT open-source optimization site containing background on algorithms https://nlopt.readthedocs.io/en/latest/ We use the information-theoretic estimator of Kitamur and Stutzer (1997). Reversing 'p' and 'p_' leads to the empirical likelihood" estimator of Qin and Lawless (1994). See Robertson et al, "Forecasting Using Relative Entropy" (2002) for more theory