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Lets say I have a number of observations and a prior. The observations are poisson distributed, are i.i.d and the prior is exponential with paramater 2. I want to calculate the maximum a posterior estimate of $\lambda$. Should I multiply every observation with the prior? Like this

$ argmax_{\lambda}\prod_{n=1}^N p(x_n \lvert \lambda)p(\lambda)=argmax \quad \lambda^{\sum_{n=1}^Nx_n}e^{-\lambda N}e^{-2 \lambda N} $

Or should I just multiply the likelihood by the prior like this

$ argmax_{\lambda}\prod_{n=1}^N p(x_n \lvert \lambda)p(\lambda)=argmax \quad \lambda^{\sum_{n=1}^Nx_n}e^{-\lambda N}e^{-2 \lambda} $

Notice the difference in the exponent on the last term

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  • $\begingroup$ No, this is a standard probability argument: the prior is the unconditional distribution/density of $\lambda$ and the likelihood is the conditional distribution/density of the sample. The product is thus the joint distribution/density of the sample and the parameter. The prior thus only appears once. $\endgroup$ – Xi'an Mar 19 at 17:31
  • $\begingroup$ @Xi'an Thanks! So it is like the second equation? $\endgroup$ – user5744148 Mar 19 at 18:52

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