Turing with both Poisson and Gaussian noise

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1

I have a set of data with both Poisson and Gaussian noise, which distribution I should choose in likelihood?
My dataset looks like

obs = data + Poisson + Gaussian

PS
My data is a series of images, it contains Gaussian and poisson noises. Iam not sure which likelihood function I should choose

2

Could you describe a bit more what you are modelling?

3

I mean for Gaussian noise, we usually use y ~ Normal(predict, sigma) to describe the noise in observation noise; But what if the noise is Poisson; and how to deal with the mixture noise?

4

I’m assuming you mean your observation model looks like

x ~ Poisson.(λ)
y ~ Normal.(m, s)
obs .= x .+ y

and you want a likelihood for obs. If your number of data points isn’t too high, you could do this:

x ~ Poisson.(λ)
obs ~ array_dist(Normal.(m .+ x, s))

which would model the Poisson-variate. The downside here is that you then can’t use NUTS because you now have discrete parameters.

If λ is large enough, Poisson is approximately Gaussian (restricted to integer support) with mean and variance λ. You could then convolve the two data distributions and use

obs ~ array_dist(Normal.(m .+ λ, s .+ sqrt.(λ)))