Multiple uncertainty corrected

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, $$Q[y_k,q_l] = M[y_k, x_i] \Pi[x_i,h_j] I[h_j, q_l]$$

$M(y_1, x_1) (\Pi(x_1,h_1)I(h_1, q_1) + \Pi(x_1,h_2)I(h_2, q_1) + \dots) + M(y_1, x_2) (\Pi(x_2,h_1)I(h_1, q_1) + \Pi(x_2,h_2)I(h_2, q_1) + \dots) +\dots$ - $$[\tilde{y}_t, y_k, q_l] = M[\tilde{y}_t, \tilde{x}_s] M[y_k, x_i] f[\tilde{x}_s, x_i, h_j] I[h_j, q_l]$$