I define a prediction of return of an asset as the following: at time $t=0$, I use my data and output that I expect the asset to make the following returns (in expected value) in the next n intervals $[r_1, r_2, \dots, r_n]$. Here, I am doing a multi-horizon forecast, i.e. at time $t=0$ I have (expected value and variance) predictions for next $n$ steps. We ignore the variance predictions for this discussion. The way $r_i$ is defined is if the asset is held from time $t=i-1$ to $t=i$, I expect it to make a return $r_i$.
Now, say I have another prediction on the same asset at time $t=0$, as $[p_1, p_2, \dots, p_m]$ But now the frequency is different. For example $n=8$, while $m=2$ but both predict return over next 48 hour period. The first predictor provides 8 predictions over the 6 hr periods, while the second provides two predictions over 24 hr periods, thus, both predicting the return profile over the next two days, but at different frequency. We call this this the prediction profile.
We are also given scalars $\pi_r$ and $\pi_p$ giving relative importance of the two prediction profiles and they sum to one, so can be taken as probability of belief.
How do we go about principally combining them into a single prediction. Please feel free to make any assumptions that make it practically suitable and closer to reality.
