# Tag Info

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Let us consider a more general case with arbitrary $\sigma$-fields $\mathcal{B}\subseteq\mathcal{A}$, $\mathcal{T}\subseteq\mathcal{S}$ and $\mathcal{M}\subseteq\mathcal{A}\otimes\mathcal{S}$. The equations $$\left(1.4.16\right)\hspace{10 mm}\Pi^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}= \mu^\mathcal{M}_\mathcal{B}\otimes ... 1 Let us consider a more general case with arbitrary \sigma-fields \mathcal{B}\subseteq\mathcal{A}, \mathcal{T}\subseteq\mathcal{S} and \mathcal{M}\subseteq\mathcal{A}\otimes\mathcal{S} and a combined conditional/marginal experiment$$ \mathcal{E}^\mathcal{M}_{\mathcal{B}\vee\mathcal{T}}=\left(A\times ...

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Bayesian and Frequentist statistics rest upon a different understanding of what a probability distribution is. To a Bayesian, probability reflects degree of belief. The purpose of an experiment (or data collection) is to update one's beliefs about the parameter values of the system. And since our beliefs concern parameters (unknown) rather than data (which ...

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######################################################################################## #These values for n, m, x, y,a, and b are just randomly chosen for illustration. These values will come from your data n = 100 m = 100 x = rbinom(n,1,.7) y = rbinom(m,1,.3) a = .2 b = .1 ...

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Just notice that your unnormalized posterior factors as $u(\mu_1,\sigma_1) v(\mu_2,\sigma_2)$, for some suitable functions $u$ and $v$, and that's enough to prove the desired independence.

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I'd not heard of the Barber book before, but having had a quick look through it, it does look very very good. Unless you've got a particular field you want to look into I'd suggest the following (some/many of which you've probably already heard of): Information theory, inference and learning algorithms, by D.J.C Mackay. A classic, and the author makes a ...

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This sounds like an application for the two-sample KS test, which evaluates whether two samples were taken from the same continuous probability distribution. For more information, you might want to start here, which explains it in some detail: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm.

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In the BUGS code, mean[i]<-inprod(X[i,],beta) should be mean[i]<-inprod(X[i,],beta[]). Your priors on tau and taubeta are too informative. You need a non-informative prior on betaifincluded, use e.g. a gamma(0.1,0.1) on taubeta. This may explain why you get tiny regression coefficients.

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A quick Google search led me to this article: Estimating the Correlation in Bivariate Normal Data With Known Variances and Small Sample Sizes. In this article, they discussed several possible priors: "uniform", "Jeffreys", and "arc-sine". Specifically, take the "uniform" prior for example, it assumes that $\rho$ follows a uniform distribution on $[-1,1]$. ...

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I think the most appropriate summary for this application is just to simply state that the years were 15% different, ± some uncertainty. Here's why: first, the concept of statistical significance per se doesn't fit naturally into the Bayesian framework; second, in this application it's not really plausible that the parameter takes the exact same value in the ...

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Expanding on my comment; let's simplify. Suppose there is one Boolean variable A. Suppose that, if the animal is a chicken, A is true 60% of the time and, if the animal is a penguin, it is true 10% of the time. Now given A, what is the probability it is a chicken? No way to know. We could have 1000 chickens - 600 with A = 1 and 400 with A = 0 10 penguins ...

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The problem arises because the negative binomial distribution can be formulated differently. As a consequence, the expectation differs for different formulations. The way you have specified the negative binomial distribution, the expectation of $n$ is $E(n) = m/\theta$ (e.g. see here on page 3). With that, the Fisher information simplifies to $$I(\theta) = ... 2 The null hypothesis isn't equivalent to a Bayesian uninformative prior for the simple reason that Bayesians can also use null hypotheses and perform hypothesis tests using Bayes' factors. If they were equivalent, Bayesians wouldn't use null hypotheses. However, both frequentist and Bayesian hypothesis testing incorporate an element of self-skepticism, in ... 0 You need to multiply f(x,y|u_i,a,b,\Sigma) and p(u_i) to get a joint distribution of f(x,y,u_i|a,b, \Sigma): f(x,y,u_i|a,b, \Sigma) = f(x,y|u_i,a,b,\Sigma) * p(u_i) = \frac{1}{\sqrt{2\pi}}exp(-\frac{1}{2}[\frac{(x-u_i)^2}{\sigma_x^2}+\frac{(y-a-bu_i)^2}{\sigma_y^2}])* \frac{1}{\sqrt{2\pi}\tau}exp(-\frac{1}{2}\frac{(u_i-\mu)^2}{\tau^2}) Note, the ... 1 As far as I know, no such book exists yet as the area is still quite new. The couple Bayesian nonparametrics books I've seen are basically just a bunch of review papers from various researchers bound together. If you have a PhD in math, applied or not, I'm sure you can get your head around by reading the standard papers. Probably the gentlest yet most ... 3 I'm the one that created the graphic, though as noted in the accompanying post it's not originally my insight. Let me provide some context for how it came up and do my best to explain how I understand it. The realization occurred during a discussion with a student who had mostly learned the Bayesian approach to inference up to that point. He was having a ... 1 This is a good reference link. If the 2 slides are not enough, the above link explains the idea and demonstrates using examples. 1 Not sure that is a helpful question: EC2 pricing is clearly described by the vendor. Relative speed to your desktop is an empirical question. Why not measure it? "Particularly suited to running occassional R and JAGS simulation" is not a questions, could you be more focused here? It obviously takes longer to run them sequentially. But then you knew that, ... 2 This is a fairly straightforward example to check for one simple reason: Your prior is the conjugate prior for Poisson data. Your prior \lambda \sim Exp(1), can be written as a Gamma distribution, because$$\lambda \sim Exp(1) \Rightarrow \lambda \sim \Gamma(1,1). The posterior distribution should then also be a Gamma. I won't go through the math on it, ...

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It depends on the specific naive model you apply. In general, for Text Classification, you do want to consider repetition of terms, so the answer is yes. The other point is that you are considering the probability based on the document event space. You can also do it based on the term space: p(viagra | spam) = 5 times spam term in class spam / 50 terms in ...

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I think it depends on what exactly you mean by p(viagra|spam) and how you're modelling the data. As written, I would interpret your meaning as 'the probability the word viagra is mentioned at least once in a message, given this message is spam'. In that case, yes, the fact that one document mentioned viagra three times has no effect. You've defined a model ...

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Without further details about your model, priors and etcetera, the best thing I can do is to point out a reference for Bayesian variable selection Faraggi and Simon (1998). Bayesian Variable Selection Method for Censored Survival Data. Biometrics Vol. 54, No. 4, pp. 1475-1485. Of course, the typical way for conducting Bayesian model selection is the ...

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From your link: Confusion of the inverse, also called the conditional probability fallacy or the inverse fallacy, is a logical fallacy whereupon a conditional probability is equivocated with its inverse i.e. this is where you make the mistake of thinking P(A|B) is the same as P(B|A). The likelihood is, however, not regarded as a conditional ...

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In general, frequentist and Bayesian estimators do not coincide, unless you use a degenerate flat prior. The main reason is this: Frequentist estimators often strive to be unbiased. For example, frequentists often try to find the minimum variance unbiased estimator (http://en.wikipedia.org/wiki/Minimum-variance_unbiased_estimator). Meanwhile, all ...

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It's not saying P(theta | data) = P (data | theta). It's defining the likelihood function.

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I am not directly familiar with Generalised Wishart Processes, but I have recently been working with the Wishart Affine Stochastic Correlation (WASC) model and hence "ordinary" Wishart processes in continuous time. The parameter corresponding to $\nu$ is usually called $\beta$, here. In fact, $\beta$ can be a real number with $\beta > n-1$ for the model ...

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See this Wikipedia article: For the case of a single parameter and data that can be summarized in a single sufficient statistic, it can be shown that the credible interval and the confidence interval will coincide if the unknown parameter is a location parameter (...) with a prior that is a uniform flat distribution (...) and also if the unknown ...

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The reason you don't have the same epiphanic look on your face as that guy is I think that . . . the statement isn't true. A null hypothesis is the hypothesis that any difference between the control and experimental conditions is due to chance. An uninformative prior is meant to state that you have prior data on a question, but that it doesn't tell you ...

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You cannot conclude the first two minutes now have "a probability density of 0" just because you did not observe an arrival in that time. If the time between arrivals is truly exponential, then there is a postive probability of an arrival in the next two minutes regardless of your single observation. Also, the distribution is memoryless, so think about what ...

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