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I am reading this article an am a bit confused about one of the terms.

My understanding is we are trying to estimate a gaussian distribution Θ from which a certain value we are interested in is drawn.

We make an initial assumption, our "prior", about such distribution and label it p(Θ).

We know (by bayes' theorem) that p(Θ|data) ~ p(data|Θ)p(Θ). Where p(Θ|data) is our updated believe after observing some real world-data about the value we are interested in.

What I don't fully get is the meaning of p(data|Θ). This to me reads as "probability of observing the real value given it is drawn from its distribution". But why would this be a probability? Is it a probability simply because we assume our measurements would be noisy/subject to stochastic?

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    $\begingroup$ Think about a simple case: flipping coins where the true probability of "heads" is unknown. Call it $\theta$. The data are the sequence of tosses such as HTHHTTTH. The likelihood of the data is $\theta (1 - \theta) \theta \theta (1 - \theta) (1 - \theta) (1 - \theta) \theta$. The probabililty of observing what we observed is this function of $\theta$. In this case we have no noise in the observation (measurement error) and it's still a probability. $\endgroup$ – Frank Harrell Jul 3 at 10:53
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It is not noise per se, but your data has a data generating process associated with it, otherwise you will just be observing a constant value for your data. It could be a normal distribution with some unknown mean $\theta$ but known variance $\sigma^2$, so that your data has an associated probability density $N(\theta, \sigma^2)$.

I would define $p(\text{data}\rvert \theta)$ as the probability of observing the data that you have observed, given a certain value of the parameter $\theta$. It is not the "real value of data" as you've stated, because our data follows a data generating process and has no one "real value". The "real value" is associated with $\theta$, and we are trying to infer this "true value" of $\theta$ based on updating our prior beliefs with the data that we observe.

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In Bayesian theory, things on right side of the equation are related to prior, meaning that they are something about your initial belief. So $p(Data|\theta)$ should read given that your initial belief of the distribution of $\theta$ (a random variable has support [0,1]), the probability of observing the specific data which is the likelihood of the data.

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