Show that every martingale 
Proceed as follows.
![\displaystyle \mbox{cov}(\theta_{d}-\theta_{c},\theta_{b}-\theta_{a}) \\ = E\left[(\theta_{d}-\theta_{c}-E(\theta_{d}-\theta_{c}))(\theta_{b}-\theta_{a}-E(\theta_{b}-\theta_{a}))\right] \\ = E\left[(\theta_{d}-\theta_{c}-E\theta_{d}+E\theta_{c})(\theta_{b}-\theta_{a}-E\theta_{b}+E\theta_{a})\right] \\ = E\left[(\theta_{d}-\theta_{c})(\theta_{b}-\theta_{a})\right]\mbox{ since }E\theta_{k}=E\theta_{1}\mbox{ for all }k=0 \\ = E\left[(\theta_{d}-E(\theta_{d}|\mathcal{D}_{c}))(\theta_{b}-E(\theta_{b}|\mathcal{D}_{a}))\right]\mbox{ by martingale definition} \\ = E\left[(E(\theta_{n}|\mathcal{D}_{d})-E(\theta_{n}|\mathcal{D}_{c}))(E(\theta_{n}|\mathcal{D}_{b})-E(\theta_{n}|\mathcal{D}_{a}))\right] \\ \mbox{ by martingale definition} \\ = E\left[E(\theta_{n}|\mathcal{D}_{d})E(\theta_{n}|\mathcal{D}_{b})\right]-E\left[E(\theta_{n}|\mathcal{D}_{b})E(\theta_{n}|\mathcal{D}_{c})\right]-E\left[E(\theta_{n}|\mathcal{D}_{d})E(\theta_{n}|\mathcal{D}_{a})\right]+E\left[E(\theta_{n}|\mathcal{D}_{c})E(\theta_{n}|\mathcal{D}_{a})\right] \\ = EE(\theta_{n}^{2}|\mathcal{D}_{d})-EE(\theta_{n}^{2}|\mathcal{D}_{c})-EE(\theta_{n}^{2}|\mathcal{D}_{d})+EE(\theta_{n}^{2}|\mathcal{D}_{c}) \\ \mbox{ because the finer decomposition subsumes the coarser one} \\ = 2\theta_{n}^{2}-2\theta_{n}^{2}\mbox{ by total probability formula} \\ = 0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmbox%7Bcov%7D%28%5Ctheta_%7Bd%7D-%5Ctheta_%7Bc%7D%2C%5Ctheta_%7Bb%7D-%5Ctheta_%7Ba%7D%29%09%5C%5C+%3D+E%5Cleft%5B%28%5Ctheta_%7Bd%7D-%5Ctheta_%7Bc%7D-E%28%5Ctheta_%7Bd%7D-%5Ctheta_%7Bc%7D%29%29%28%5Ctheta_%7Bb%7D-%5Ctheta_%7Ba%7D-E%28%5Ctheta_%7Bb%7D-%5Ctheta_%7Ba%7D%29%29%5Cright%5D+%5C%5C+%3D+E%5Cleft%5B%28%5Ctheta_%7Bd%7D-%5Ctheta_%7Bc%7D-E%5Ctheta_%7Bd%7D%2BE%5Ctheta_%7Bc%7D%29%28%5Ctheta_%7Bb%7D-%5Ctheta_%7Ba%7D-E%5Ctheta_%7Bb%7D%2BE%5Ctheta_%7Ba%7D%29%5Cright%5D+%5C%5C+%3D+E%5Cleft%5B%28%5Ctheta_%7Bd%7D-%5Ctheta_%7Bc%7D%29%28%5Ctheta_%7Bb%7D-%5Ctheta_%7Ba%7D%29%5Cright%5D%5Cmbox%7B+since+%7DE%5Ctheta_%7Bk%7D%3DE%5Ctheta_%7B1%7D%5Cmbox%7B+for+all+%7Dk%3D0+%5C%5C+%3D+E%5Cleft%5B%28%5Ctheta_%7Bd%7D-E%28%5Ctheta_%7Bd%7D%7C%5Cmathcal%7BD%7D_%7Bc%7D%29%29%28%5Ctheta_%7Bb%7D-E%28%5Ctheta_%7Bb%7D%7C%5Cmathcal%7BD%7D_%7Ba%7D%29%29%5Cright%5D%5Cmbox%7B+by+martingale+definition%7D+%5C%5C+%3D+E%5Cleft%5B%28E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bd%7D%29-E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bc%7D%29%29%28E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bb%7D%29-E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Ba%7D%29%29%5Cright%5D+%5C%5C+%5Cmbox%7B+by+martingale+definition%7D+%5C%5C+%3D+E%5Cleft%5BE%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bd%7D%29E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bb%7D%29%5Cright%5D-E%5Cleft%5BE%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bb%7D%29E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bc%7D%29%5Cright%5D-E%5Cleft%5BE%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bd%7D%29E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Ba%7D%29%5Cright%5D%2BE%5Cleft%5BE%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Bc%7D%29E%28%5Ctheta_%7Bn%7D%7C%5Cmathcal%7BD%7D_%7Ba%7D%29%5Cright%5D+%5C%5C+%3D+EE%28%5Ctheta_%7Bn%7D%5E%7B2%7D%7C%5Cmathcal%7BD%7D_%7Bd%7D%29-EE%28%5Ctheta_%7Bn%7D%5E%7B2%7D%7C%5Cmathcal%7BD%7D_%7Bc%7D%29-EE%28%5Ctheta_%7Bn%7D%5E%7B2%7D%7C%5Cmathcal%7BD%7D_%7Bd%7D%29%2BEE%28%5Ctheta_%7Bn%7D%5E%7B2%7D%7C%5Cmathcal%7BD%7D_%7Bc%7D%29+%5C%5C+%5Cmbox%7B+because+the+finer+decomposition+subsumes+the+coarser+one%7D+%5C%5C+%3D+2%5Ctheta_%7Bn%7D%5E%7B2%7D-2%5Ctheta_%7Bn%7D%5E%7B2%7D%5Cmbox%7B+by+total+probability+formula%7D+%5C%5C+%3D+0+&bg=ffffff&fg=333333&s=0&c=20201002)
As a quick note as to why 
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Martingales – Problem (29/365)
Show that every martingale
Proceed as follows.
As a quick note as to why
Latent observations 