I would like to define a log-likelihood (starting with a gaussian distribution) for an observed value of a quantity ( ![enter image description here][1] ), compared to the measured value based on a given model ( ![enter image description here][2] ) and instead of using the measurement errors for each observation ( ![enter image description here][3] ), I would like to use the weighing value which is a combination of errors and another measured parameter, i.e. ![enter image description here][4], where ![enter image description here][5] can be a given constant value. It is easy to show that ![enter image description here][6] 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?


  [1]: https://i.sstatic.net/UfhV8.gif
  [2]: https://i.sstatic.net/7sqGe.gif
  [3]: https://i.sstatic.net/qYUCZ.gif
  [4]: https://i.sstatic.net/FoVPh.gif
  [5]: https://i.sstatic.net/qY956.gif
  [6]: https://i.sstatic.net/0gnXM.gif