weighing the maximum likelihood

Question

I would like to define a log-likelihood (starting with a gaussian distribution) for an observed value of a quantity ($x_i$), compared to the measured value based on a given model ($\hat{x_i}$) and instead of using the measurement errors for each observation ($\sigma_i$), I would like to use the weighing value which is a combination of errors and another measured parameter, i.e. $w_i = 1/(\sigma_i^2+\alpha)$, where $\alpha$ can be a given constant value. It is easy to show that $w_i$ has reverse property as error, meaning it is higher for values with smaller errors mostly.

How could I re-write the likelihood and use the weight value for each measurement instead of errors?

Answer 1

It sounds like you have observations comprising normal distribution with parameters $x, \sigma^2 + \alpha$. You have model predictions $\hat x, \hat \sigma^2$. Typically, when people don't mention $\hat \sigma^2$, they're just fixing it at one. Then, rather than a log-likelihood, you want its generalization, the cross entropy.

It also sounds like you were thinking of weighting the components of the log-likelihood (or cross entropy) by some $w_i$. Do this only if $w_i$ represents missingness, which it doesn't in your definition.