I would like to define a log-likelihood (starting with a gaussian distribution) for an observed value of a quantity ( ), compared to the measured value based on a given model ( ) and instead of using the measurement errors for each observation ( ), I would like to use the weighing value which is a combination of errors and another measured parameter, i.e. , where can be a given constant value. It is easy to show that 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?