Fixed the multiple uncertainty calculation implementation. Code had transposed \(\mathbb{I}\), also one of the \(\mathbb{M}\) matrices. Should still confirm final implementation. In the below example each case has log-normal growth noise present. The non-monotonic section of the measurement uncertainty
See notes on code changes inline.
Additional notes from walking through algoritm step by step.
- \(x\) index in the true state space
- \(y\) index in the observed state space
- \(h\) index in the true action
\(q\) index in the policy/decision space
- \(V[y,q]\) the Expected value of policy \(q\) and observed state \(y\)
- \(\Pi[x,h]\) the profit of a harvest of \(h\) given a true stock of \(x\)
- \(M[y,x]\) is the probability of a true state \(x\) given an observed stock \(y\). (How does this relate to the probability the observed stock is \(y\) given the true stock is \(x\)? Should be the transpose? Is \(M\) symmetric?)
- \(I[h,q]\) is the probability of implementing a harvest \(h\), given a quota set to \(q\).
\(Q[y,q]\) the Expected profit from a policy \(q\) and observed state \(y\). In Einstein summation, \(Q[y_k,q_l] = M[y_k, x_i] \Pi[x_i,h_j] I[h_j, q_l]\)
\[M(y_1, x_1) (\Pi(x_1,h_1)I(h_1, q_1) + \Pi(x_1,h_2)I(h_2, q_1) + \dots) + M(y_1, x_2) (\Pi(x_2,h_1)I(h_1, q_1) + \Pi(x_2,h_2)I(h_2, q_1) + \dots) +\dots\] - \([\tilde{y}_t, y_k, q_l] = M[\tilde{y}_t, \tilde{x}_s] M[y_k, x_i] f[\tilde{x}_s, x_i, h_j] I[h_j, q_l]\)