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I am afraid this will be quite obvious once you tell me, but I'm just confused right now.

Say we have an i.i.d. RV $N\sim Poisson(\mu)$. Now, I want to calculate the likelihood $L$ of an observation $D = \{n_1,n_2\}$. As I expected, resources seem to suggest $$L=Poisson(n_1,\mu)*Poisson(n_2,\mu).$$ But, why wouldn't it be twice as much i.e. $p("n_1~then~n_2") +p("n_2~then~n_1")$ ?

I seem to somehow (unneccesarily) introduce an order of the observations... can someone give me a formal explanation why the Likelihood is what it is?

(Related question; How would the likelihood be calculated if the order matters?)

bebissig
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1 Answers1

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Ordering doesn't really apply to likelihood functions. For your likelihood, you calculate the probability that your variable N would take on the value $n_1$ and $n_2$ under certain parameters $\mu$. You can see the dataset $D$ as an ordered realisation of a random sample $N_1,N_2$ from $N$, and so $N_1 = n_1$ and $N_2=n_2$.

This would give you $P_\mu(N_1=n_1,N_2=n_2) = P_\mu(N_1=n_1)P_\mu(N_2=n_2) = P_\mu(N_2=n_2)P_\mu(N_1=n_1)$, where of course the order in which you multiply your probabilities doesn't really matter.

mariekejee
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  • If I read your first sentence as a defining property of likelihood, then maybe ok. Still I could ask what is likelihood in two coin flips for observing (1xH,1xT) or what is likelihood for (H then T) (in order). I still think these are two different numbers (0.5 vs. 0.25). I am asking this coming from the calculation of the likelihood in the Bayesian numerator. I could imagine, that in that case the "likelihood" in numinator and denominator have to be the same, hence the binomial coefficient would cancel and in fact it does'nt matter which likelihood i choose. However, is this generally true? – bebissig Jun 27 '19 at 11:40
  • @bebissig What you've described is a probability (see [here](https://stats.stackexchange.com/questions/2641/what-is-the-difference-between-likelihood-and-probability)). But I see your point. Depending on what you know of the outcome, this likelihood indeed changes. But as you also noted, this would be incorperated in both the numerator and denominator, and so the constant is irrelevant. Many texts just use $\propto$ symbol because the constants are only there to ensure the prior is a probability distribution (i.e. integrates to 1). Same with mle, where the constant has no impact. – mariekejee Jun 27 '19 at 12:56
  • as far as i see in e.g in the secomd answer, likelihood and prob seem to be identical, only one is func of param. while the other of the data. i was just confused not to find a very explixit definition. and, e.g pymc gives the (log) ordered probability when querying .logp of an observed stochastic variable. i'll leave this question open bit longer, maybe someone can tell. – bebissig Jun 28 '19 at 03:27