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Extended comment. In slide 4/6 of the lecture slides which you linked in the comments: In light of this, I am unable to understand how or why your query remains concerning whether the posterior is a Gaussian process? Is the presence of $\mathbf{x}$ and the $\mathcal{M}_i$ on the right hand side preventing you from recognising the $p(f | \mathbf{x}, \mathbf{...


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You have mixed a couple of ideas. A $t$-test is a parametric test (although the Bayesian $t$-test can relax the assumptions) while the Wilcoxon test treats Y as ordinal and uses only its rank ordering. To get a Wilcoxon test, use its generalization the proportional odds ordinal logistic regression model. The blrm function in the R rmsb package makes this ...


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As pointed out by others, this rule derives from the statement by the famous English General Oliver Cromwell, who stated: "I beseech you, in the bowels of Christ, think it possible that you may be mistaken." Applied to a the context of Bayesian analysis, it recommends generally against assigning a prior probability of zero or one to an unknown ...


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According to Wikipedia, the rule states that: the use of prior probabilities of 1 ("the event will definitely occur") or 0 ("the event will definitely not occur") should be avoided, except when applied to statements that are logically true or false, such as 2+2 equaling 4 or 5. The motivation for the rule is that if you assign a prior ...


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Okay I got this one myself. $X\sim Poisson(\lambda), \pi(\lambda)=Gamma(1,1/k)$. The posterior is obtained as: $$\pi(\lambda|x)\propto p(x|\lambda)\pi(\lambda)\propto\lambda^xe^{-\lambda}e^{-\lambda/k}\propto\lambda^xe^{-(\lambda(1+1/k))}$$ $$\lambda|x\sim Gamma(X+1,1+1/k)$$ Bayes rule: $$E[\frac{(\lambda-a)^2}{\lambda}|X]=a^2E(1/\lambda|X)-2a+E(\lambda|X)$$ ...


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Prior effective sample size (ESS) doesn't have a single definition. As far as I can tell, it's a heuristic to understand the influence of the prior on the posterior parameters. Prior ESS tells your choice of prior is comparable to having an additional $n_{E}$ data points. It is straightforward to demonstrate prior ESS with conjugate priors. It is more ...


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Previous experiment. Suppose data and two-sample t test for the previous experiment were as follows: summary(x1); length(x1); sd(x1) Min. 1st Qu. Median Mean 3rd Qu. Max. 46.07 47.71 49.93 49.81 51.80 53.46 [1] 40 [1] 2.284828 summary(x2); length(x2); sd(x2) Min. 1st Qu. Median Mean 3rd Qu. Max. 47.49 50.33 ...


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Here is an example with a beta prior distribution and a binomial likelihood. Suppose the prior distribution of the heads probability $\theta$ of a coin is $\mathsf{Beta}(10,10)$ and that $n = 100$ tosses of the coin yield $x = 47$ Heads. Then the posterior distribution of $\theta$ is $\mathsf{Beta}(10 + x, 10 + 100 - x) \equiv \mathsf{Beta}(57, 63).$ This ...


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I'm not clear on the question, so I'm going to write some exposition and then finish my answer when you've clarified. The Maximum A Posteriori estimate is, as you've said, is $$ \hat{\lambda} = \underset{\lambda \in \mathbb{R}_+}{\mbox{argmax}} \left\{ \ell(\lambda; x) + \log(p(\lambda)) \right\} $$ Here, $\ell(\lambda;x)$ is the log likelihood and $p(\...


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