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Dalek
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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?

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?

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?

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Hong Ooi
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I would like to define a log-likelihood (starting with a gaussian distribution) for an observed value of a quantity ( enter image description here $x_i$), compared to the measured value based on a given model ( enter image description here $\hat{x_i}$) and instead of using the measurement errors for each observation ( enter image description here $\sigma_i$), I would like to use the weighing value which is a combination of errors and another measured parameter, i.e. enter image description here$w_i = 1/(\sigma_i^2+\alpha)$, where enter image description here$\alpha$ can be a given constant value. It is easy to show that enter image description here$w_i$ 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?

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

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?

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

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

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

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Dalek
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