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great to see you are interested in GPs. Someone asked a similar question about how to make predictions on a GP and I think I gave a fairly comprehensive work through with sample data that they gave. I think seeing it in action will answer your question. Here is the link.


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The condition that a probability density must integrate to 1 gives you that its normally distributed.


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The three cards in your friend's hand and burned don't matter unless the betting tells you something about the chance your friend has a Jack. On the turn, you have two favorable outcomes out of $47$ because you know five of the cards. Assuming you don't get a Jack on the river the chance you have two favorable out of $46$ cards for the river. The chance ...


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This is one of the basic steps in the Expectation Propagation algorithm. See section 3.2 of A family of algorithms for approximate Bayesian inference, where the formulas you want are given by (3.32,3.33). Keep in mind that the variance of the likelihood might turn out to be negative, so SD2 would be imaginary. This is why it is better to write the ...


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These quantities cannot be computed without knowing the missing entries of the conditional probability tables.


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To compute the MLE of the parameters, you need to marginalize out the latent variables $(X,Z)$ which is difficult. However, MCMC does not require this. With MCMC, you can jointly sample the parameters and $(X,Z)$, then ignore the latter.


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No, you cannot make a stronger conclusion. That is because Bayes nets are "I-maps" (in Pearl's terminology): d-separation implies independence but independence does not imply d-separation. See section 3.1.3 of Probabilistic Reasoning in Intelligent Systems.


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ok here is an attempt What you are looking for in general are Bayesian hierarchical models. A good reference for all things Bayesian is Gelmans Bayesian Data Analysis. For your example this is a hierarchical model, which follows this logic 1) Define your likelihood $y_{ij}|\alpha_i,sigma^2,\mu ~ N(\mu + \alpha_i,\sigma^2_{ij})$ In this model you are ...


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It implies that $y$ is uniform over its domain. Since density is 1/area, it also implies that $y$'s domain has area 1/5.


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A good way to understand how these two perspectives differ is to look at how they approach the problem of confidence intervals. The question What's the difference between a confidence interval and a credible interval? has some excellent discussion on this, including examples where both approaches are taken and shown to give different answers.


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Can I ask about what is the definition of "proper" or "improper"? Actually I also encountered the expression "improper uniform distribution" while reading a paper, but couldn't find the definition of it. According to your question, "improper distribution" looks a distribution whose integration on support is not equal to 1. Is it right?


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What we want to show is that $E\left[Var(\omega|X_{1},..,X_{n})\right]\leq Var(w)$. Well by the law of total variance we have that $$Var(w)=E\left[Var(\omega|X_{1},..,X_{n})\right]+Var(E[\omega|X_{1},..,X_{n}])$$ and since $Var(Y)\geq 0$ for any rv $Y$. $Var(E[\omega|X_{1},..,X_{n}])\geq 0$ we have that $$\leq E\left[Var(\omega|X_{1},..,X_{n})\right]$$


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If $N$ and $M$ are large, you expect the first part to be a good measure of $\theta$, then you expect a fraction $\theta$ of the tosses to be heads, so your prediction should be $\theta M$, strategy 2. If they are large enough that the normal approximation is good, you can use that. All heads will then be several standard deviations away. If $N,M$ are ...


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For intuition, consider how you'd measure the frequency by eye, if you don't have a lot of computational resources. Simply counting waves and troughs gives you a fairly good initial estimate of the frequency. In order to make it more precise, what you'd want to do is find two zero crossings and divide the number of waves between them with their distance. ...


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FYI, using the chain rule, P(a,z,b) = P(a|z,b)*P(z|b)*P(b)


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We can look to Bayes formula for inspiration. It can be derived from the definition of the joint distribution: $$P(A, B) = P(A|B) \,P(B) = P(B|A)\,P(A)$$ and rearraning to give $$P(B|A) = \frac{P(A|B) \,\,P(B)}{P(A)}$$ For the case of 4 variables, we have many more options. Below is one example of a formula Example We could write the joint distribution ...


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It seems that your parameter $\theta\sim\Gamma(\alpha,\beta)$ and observations $(X_1,X_2,\ldots,X_n)\sim P(\theta)$. So prior density $\pi(\theta)$=density of $\Gamma(\alpha,\beta)$ and p.d.f of the random vector $P(x_1,x_2,\ldots,x_n\setminus \theta)=P(\theta)$. Therefore, posterior density is =$f(\theta\setminus ...


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Hi if your process is Gaussian then the joint law $(X_t,X_{t+\tau})$ is a multivariate Gaussian random variable. You only need to apply in this case the formulas in wiki link at the section on conditional distribution to get what you want. Best regards



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