Suppose I have a log-likelihood of the form $$\mathcal{L} = \sum_{i = 1}^{n} a_i + \sum_{j = 1}^{m} b_j,$$ where $a_i$ and $b_j$ are some independent log-probabilities. The problem is, the second sum $\sum b_j$ has much smaller contribution than the first one ($m \ll n$ and $|a_i|$ is usually bigger than $|b_j|$).
Obviously, the second summation can be multiplied by some weight $C$, but I have several questions:
- Are there any methods to find weight $C$ in advance? (sizes $n,m$ and distributions $a,b$ are known),
- Are there different techniques to equalize contribution of different terms in log-likelihood?