# Tag Info

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"This scenario" possibly refers to one of two things: a test of whether Y is associated with X (with no assumptions of their functional relationship) or a test of whether that relationship is linear. The Bayesian framework is much better for arguing in favor of one model versus another, so this seems adequately suited to the latter problem. It's worth ...

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Since GAM are nonparametric models, you will need to look at the literature on Bayesian Nonparametrics in order to find analogous models. This theory might be slightly more difficult to digest, though, given that the priors have to set on infinite dimensional spaces. If you are brave enough to dig into this area, I would recommend the following book as a ...

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As stated in comment, the prior distribution represents your prior beliefs about the distribution of the parameters. When you have prior beliefs you can: convert your belief in terms of moments (e.g. mean and variance) to fit a common distribution to these moments (e.g. Gaussian if your parameter lies to the real line) use your intuitive understanding ...

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Giving the author and title of the book would be helpful in deciphering the author's intention, since readers here might have read it. But based on this information, it would appear to simply be a teaching approach intended to simplify the problem for illustrative purposes. Rather than estimating both parameters at once, the author estimates them in turn ...

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Yes: $$p(\theta| X) \propto p(X | \theta) p(\theta)$$ Yes. You can create any model that you want. The question is that, is that a good model for the process you model or not?

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You are right that the paper is saying the wrong thing. You certainly can evaluate the posterior distribution of $x$ at a known location using $O(n)$ operations. The problem is when you want to compute moments of the posterior. To compute the posterior mean of $x$ exactly, you would need $2^N$ operations. This is the problem that the paper is trying to ...

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The formula on that slide was a straw man and not intended to make sense. The point was that moment matching does not make sense on an individual likelihood term in isolation. This is illustrated further on the next slides. I have actually seen this bad approach used in papers, so I thought it was worth pointing out. This is one of those cases where "you ...

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The problem was that I was setting the degrees of freedom too low-- it should be at least P+2, where $\Psi$ is a PxP matrix.

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Yes. The first chapter talks mostly about support vector machines and hyperplane classifiers(among other things). So, yes, they have discussed a frequentist approach. They do discuss bayesian methods in chapter 16 and give a good explanation for the connection between regularized loss minimization and bayesian methods.

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How to integrate "prior knowledge" about the sun stability in the frequentist methodology? Very interesting topic. Here are just some thoughts, not a perfect analysis... Using the Bayesian approach with a noninformative prior typically provides a statistical inference comparable to the frequentist one. Why does the Bayesian has a strong prior ...

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Basically, the error is that a conclusion can't be drawn from $n$ = 1.

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As you stated, in some ways, the Bayesian inference solves your problem. However, it has to be considered that, roughly speaking, the less data, the more the priors speak. So in the presence of very little data, the prior distributions would have more influence, and consequently your inference can be highly conditioned by the choice of prior. In that sense, ...

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I make the potentially false assumption that fitting a probability model in the frequentist way is virtually the same as fitting the same model with a flat prior in a Bayesian way. Please nuance or correct that as interest number one (1). If the flat prior contains the Maximum Likelihood Estimator (MLE), then the MAP (Maximum A Posteriori) and the MLE ...

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I had exactly the same with specifying dgamma(0.001, 0.001) for WinBUGS! Jags can actually take it (and I recommend you to use Jags if you don't need advanced WinBUGS features; and even if you want to use WinBUGS, it comes in handy for debugging, because Jags has much more comprehensible error reporting), but for WinBUGS, you must do: x[..] ~ ...

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This is an implementation problem since, theoretically, MCMC has no problem with truncated distributions. Let $D$ be the support of your posterior. Just define your log-posterior as $-\infty$ if $\theta\not\in D$ and choose a suitable value on $D$ as an initial point. For example, suppose that $x_1,\dots,x_n \stackrel{ind.}{\sim} \exp(\lambda)$ and that ...

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Given that the two-sample Kolmogorov-Smirnov test is a nonparametric procedure, you have to dig into the area of Bayesian nonparametrics to find Bayesian analogous tools. Here is a paper where you can start: http://arxiv.org/abs/0910.5060 I warn you that nonparametric Bayesian theory is not easy to digest.

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An alternative to using a Half-Cauchy distribution with a well-defined variance is a Half-Student-t with $\nu>2$ degrees of freedom, e.g. $\nu=3$. $$\pi(\nu)= \frac{12 \sqrt{3}}{\pi \left(x^2+3\right)^2},\,\,\, \nu>0.$$ This prior has semi-heavy tails and it should produce fairly similar results as the Half-Cauchy prior. You can visualise it in R ...

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You can find the explanations on how to derive the energy function in the following link: http://web.engr.illinois.edu/~khashab2/learn/ep.pdf See section "EP energy function"

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I am responding to a request for alternative graphical techniques that show how well simulated failure events match observed failure events. The question arose in "Probabilistic Programming and Bayesian Methods for Hackers " found here. Here's my graphical approach: Code found here.

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There's no reason Jeffreys's prior is "supposed" to give you "good" results. The chief motivation behind it is that posterior inferences are invariant to reparametieriation of the model. Is that a major concern for your problem? I can't say. But if it is not, then an alternative choice could improve posterior inferences. In your case, what these results ...

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You could try the approach recommended by Steve Goodman and calculate the minimum bayes factor: Toward Evidence Based Medical Statistics 2: The Bayes Factor To get this from mcmc results, you can subtract the estimate for the group level parameters for each step to get a posterior distribution of the difference as was done by John Kruschke in this paper: ...

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A simple way to do this is to subtract the "centering value" of the coefficient times its associated variable from the left-hand side. To go with your example, $Y = \beta_1X_1 + \beta_2X_2 + \beta_3X_3 + \beta_4X_4 + e$ Assume the coefficient values should be centered at (5,1,-1,-5) respectively. Then: $Y - 5X_1 -X_2 +X_3 +5X_4 = (\beta_1-5)X_1 + ... 0 Ridge regression sets a normal prior centered at zero for each parameter. To penalize coefficients towards different values, just center the priors around your target instead of around 0. I think you can accomplish this using the bayesGLM function of the arm package with the following parameters: prior.mean : set this to vector containing the values you ... 1 For your first point: Not sure what you mean by reconciled, as I think Bayesians do not challenge the CLT. The use of the Gamma appears fine...note that the Gamma$\rightarrow$Normal as the mean approaches$\infty$. For a Bayesian analysis, your prior should exclude parameter values that cannot happen, hence they are correct than an unbounded normal would ... 1 Roughly speaking, the Bayesian and the frequentist perspectives agree on calculation of the likelihood, but disagree on the interpretation. That's because the Bayesian interpretation of probability is, very roughly speaking, a superset of the frequentist one. From the frequentist perspective, the probability only makes sense if the likelihood is describing ... 5 Fortuitous timing, as Bayesian Data Analysis, 3rd ed was just released. It's a good general-purpose text, with an emphasis on hierarchical methods, a section on advanced computation (that is, Markov chain Monte Carlo), and an appendix on Gelman's Bayesian inference tool, rstan. The text focuses on statistics rather than programming, though, so perhaps ... 9 Peter D. Huff. A First Course in Bayesian Statistical Methods. Springer (2010) Also Andrew Gelman et. al. Bayesian Data Analysis (3rd ed.). CRC (2013) The Gelman book isn't constrained to R but also uses Stan, a probabilistic programming language similar to BUGS or JAGS. I believe earlier editions of the book used BUGS instead of Stan, which is ... 1 Update: I misunderstood what was the nature of the prior information. You have a problem with your experiment, because we can't disentangle the effect of the nationality from the effect of the banner. As far as I understood, banner_past is different from banner A and B. So, you know that, on banner-past, french are slightly more likely to click on the ... 1 I believe that the problem is with your guess that the inverse gamma is so easily extended to the multivariate case. From the distributions appendix of Gleman et al, Bayesian Data Analysis (3rd Edition), 582 The Inverse-Wishart distribution is the conjugate prior distribution for the multivariate normal co-variance matrix. ... The Wishart ... 0 It is hard for me to tell exactly what you are doing without example data, but here is an example from John Kruschke. His approach may be superior because you model the strength of the autoregression in an unusual way (not saying it is wrong just not what I have seen done, which isn't that much): model { trend[1] <- beta0 + beta1 * x[1] + amp * ... 0 This has generated some interesting debate, but note that it really doesn't make much difference to the question of interest. Personally I think that because$\theta$is a scale parameter, the tansformation group argument is appropriate, leading to a prior of$\$\begin{array}& ...

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Other answered is correct about BUGS but his answer does not apply to JAGS (at least, not to rjags, R2jags might be different). I haven't used JAGS directly, but the writer of rjags is the creator of JAGS so I would guess they use the same convention. In rjags, the jags.model object keeps track of the number of iterations that the chain(s) have been run. ...

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