An apparent disagreement between two transfer formulas often comes from acting on different objects. Throughout this guide, probabilities are columns and observables are functions written as columns. Their pairing is \(f^{\mathsf T}p\). The transpose distinguishes the forward action on distributions from the backward action on observations.
| Object | Exact meaning | What to check |
|---|---|---|
| \(p\) | A nonnegative column with \({\mathbf{1}}^{\mathsf T}p=1\). | It is not a vector of function values. |
| \(Q\) | Continuous Markov generator. | Off-diagonal entries are nonnegative and columns sum to zero. |
| \(K\) | One discrete stochastic step. | \({\mathbf{1}}^{\mathsf T}K={\mathbf{1}}^{\mathsf T}\) and \(K_{ij}\ge0\). |
| \(K^{\mathsf T}f\) | Expected next observable given the current state. | The same dynamics, with the opposite action on the dual object. |
| \(P_a\) | A coordinate packet projector. | All packets together reconstruct the full state. |
| \(C\) | Sums fine probability masses into coarse blocks. | Usually rectangular and not invertible. |
| \(E_\pi\) | Conditional expectation of observables. | A weighted average, self-adjoint for the \(\pi\) inner product. |
| \(\mathcal B_\lambda\) | \(\operatorname{diag}(\lambda)p/(\lambda^{\mathsf T}p)\). | Nonnegative likelihood and positive total evidence. |
| \(L\) | A nonnegative self-adjoint operator. | Its evolution is \(e^{-tL}\), not \(e^{tL}\). |
| \(R(z)\) | \((zI-A)^{-1}\). | \(z\) belongs to the resolvent set. |
| \(\mathcal K(t)\) | The memory kernel after hidden-state elimination. | It is not the transition matrix \(K\). |
One convention test
For
we obtain \(K_2p=(3/4,1/4)^{\mathsf T}\) and
With departure weights \(w_i=e^{-V(i)}\), the backward Feynman–Kac operator is \(\operatorname{diag}(w)K_2^{\mathsf T}\). Its forward dual is \(K_2\operatorname{diag}(w)\). Interchanging these factors changes when the weight is applied.
Read this as
Always name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.