If you have several $x_{ij}$ forecasts ($j$-th providers forecast for the $y_i$ value) and want to combine them, then you are thinking about using some kind of weighted mean of them (see here for examples), e.g.
$$ y_i = \sum_j w_j x_{ij} $$
where $w_j$ are weights such that $\sum_j w_j = 1$. Weights $w_j$ in this case can be interpreted as $j$-th providers reliability.
Your model seems to be more complicated than this since you also want to include information about forecasts time distance $t_j$. First you need to notice that you are not interested in actual $y_i$'s but rather in providers errors. For this, let's define variable $z_{ij} = y_i - x_{ij}$, for difference between actual value of $y_i$ and prediction of $j$-th provider. Such difference depends on providers reliability $\alpha_j$ and the forecast delay $t_{ij}$, i.e.
$$ z_{ij} = \alpha_j + \beta t_{ij} + \varepsilon_{ij} $$
$\alpha_j$ is a random variable with mean $\bar \alpha_j$ (additive bias), and variance $\sigma_{\alpha_j}^2$ (how much forecasts by $j$-th provider vary in their error). In this particular case I assumed that the effect of delay $\beta$ is the same for all providers, but you can make different assumptions. Knowing values of those parameters you would be able to predict given the data the expected error for $j$-th provider that made his forecast delayed by $t_{ij}$ time points. You can use such bias estimates for bias correction for individual forecasts: $\hat y_i = x_{ij} + \hat z_{ij}$.
The absolute value of inverted estimated error $|\hat z_{ij}^{-1}|$ could instead of bias correction be used to weight different forecasts and make weighted forecast
$$ \hat y_i = \frac{\sum_j |\hat z_{ij}^{-1}| x_{ij}}{ \sum_j{|\hat z_{ij}^{-1}|}} $$
Since each combined forecast is weighted by expected, inverted error (the more biased it is, the less weight it has), you could expect such forecast to be (on average) better than individual forecasts.
In Bayesian setting you can set up some priors on $\alpha_j$ and $\beta$ to estimate them and then, in sequential analysis, when new data arrives use their posterior estimates as priors for new estimation (check Bayesian updating).
