agent’s expectations for different experiments. Suppose our agent has a preferred reference process consisting of a measurement to which she ascribes the MIC $\{H_i\}$, and, upon obtaining outcome $i$, the preparation of a state $\sigma_i$, drawn from a linearly independent set of post-measurement states $\{\sigma_i\}$. (See Fig. \[figure1\].) In her choice of this reference process, she requires linearly independent post-measurement states so that the inner products $\tr D_j\sigma_i$ will uniquely characterize the operators $D_j$. Let $P(H_i)$ be her probabilities for the measurement $\{H_i\}$ and $P(D_j|H_i)$ be her conditional probabilities for a subsequent measurement of $\{D_j\}$. What consistency requirement among $Q(D_j)$, $P(H_i)$, and $P(D_j|H_i)$ does quantum physics entail?

Using the fact that $\{\sigma_i\}$ is a basis, we may write $$\rho=\sum_j\alpha_j\sigma_j\;,$$ for some set of real coefficients $\alpha_j$. The probability of outcome $H_i$ is then $$P(H_i)=\sum_j\alpha_j\,\tr H_i\sigma_j=\sum_j\,\big[\Phi^{-1}\big]_{ij}\,\alpha_j\;,$$ where we have defined the matrix $\Phi$ via its inverse, $$\label{phiinv}
    \big[\Phi^{-1}\big]_{ij}:=\tr H_i\sigma_j=h_i\tr\rho_i\sigma_j\;,$$ for $\rho_i:=H_i/h_i$ and $h_i:=\tr H_i$. The invertibility of $\Phi$ is assured by the linear independence of the MIC and post-measurement sets. This implies that the coefficients of $\rho$ in the $\sigma_i$ basis may be written as an application of the $\Phi$ matrix on the vector of probabilities, $$\label{rhoinbasis}
    \rho=\sum_i\left[\sum_k[\Phi]_{ik}P(H_k)\right]\! \sigma_i\;.$$ The probability 