# Tag Info

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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 ...

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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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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 ...

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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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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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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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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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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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MCMC is a strategy for generating samples $x(i)$ while exploring the state space $X$using a Markov chain mechanism. These are irreducible and aperiodic Markov chains that have $P_{target}(\theta)$ as the invariant distribution. This mechanism is constructed so that the chain spends more time in the most important regions. In particular, it is constructed ...

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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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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: $Y - 5X_1 -X_2 +X_3 +5X_4 = (\beta_1-5)X_1 + ... 1 For your first point: Not sure what you mean by reconciled, as I think Bayesians do not challenge the CLT. The use of the Gamma appears fine...note that the Gamma$\rightarrow$Normal as the mean approaches$\infty$. For a Bayesian analysis, your prior should exclude parameter values that cannot happen, hence they are correct than an unbounded normal would ... 1 I believe that the problem is with your guess that the inverse gamma is so easily extended to the multivariate case. From the distributions appendix of Gleman et al, Bayesian Data Analysis (3rd Edition), 582 The Inverse-Wishart distribution is the conjugate prior distribution for the multivariate normal co-variance matrix. ... The Wishart ... 1 Update: I misunderstood what was the nature of the prior information. You have a problem with your experiment, because we can't disentangle the effect of the nationality from the effect of the banner. As far as I understood, banner_past is different from banner A and B. So, you know that, on banner-past, french are slightly more likely to click on the ... 1 Roughly speaking, the Bayesian and the frequentist perspectives agree on calculation of the likelihood, but disagree on the interpretation. That's because the Bayesian interpretation of probability is, very roughly speaking, a superset of the frequentist one. From the frequentist perspective, the probability only makes sense if the likelihood is describing ... 1 This is an implementation problem since, theoretically, MCMC has no problem with truncated distributions. Let$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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