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Bayesian inference is a method of statistical inference that relies on treating the model parameters as random variables and applying Bayes' theorem to deduce subjective probability statements about the parameters or hypotheses, conditional on the observed dataset.

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Bayesian hypothesis testing is usually done by formulating a model that decomposes the prior into the null and alternative cases, which leads to a particular form for Bayes factor. For an arbitrary … prior-to-posterior probability mapping, which is a bit like an AUC curve. Here is a general model form for your problem, plus the more specific uniform-prior model. General Bayesian model: Suppose …
answered Aug 26 '18 by Ben
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There is no particular reason to create a reversal of Bayes' theorem in this case, since specification of a statistical model is usually a direct specification of the sampling distribution (i.e., the …
answered May 31 '18 by Ben
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Your question is presently ill-posed, and it really depends on what kind of "aggregation" you have in mind. Each Bayesian has a legitimate posterior estimate based on observed data and a prior … estimated probabilities of the event may differ.) Now, suppose that your Bayesians have priors $\pi_1,...,\pi_N$. Then for Bayesian $i$ the posterior probability of the event of interest is: $$H_i …
answered Sep 2 '18 by Ben
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From the first line of the Wikipedia page: ... a Bayes estimator or a Bayes action is an estimator or decision rule that minimizes the posterior expected value of a loss function (i.e., the poster …
answered Oct 15 '18 by Ben
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The answer by Taylor is excellent, and already correctly gives the posterior kernel, which gives an intractable integral. If your goal is to find the posterior density (as opposed to sampling from th …
answered Dec 6 '18 by Ben
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more different subjects could all hold different beliefs, and none would be considered more or less wrong than the others). In Bayesian analysis it is generally the case that the chosen sampling … sample data. This gives rise to three broad paradigms in Bayesian statistics, which correspond to different ways of determining the prior distribution. Subjective Bayesian paradigm: This paradigm …
answered Jan 10 by Ben
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Given that $\theta_{\text{MLE}}$ is a point estimator, the obvious counterpart in Bayesian analysis would be the posterior mode estimator $\theta_{\text{MAP}} \equiv \arg \max_\theta f( \theta | x … )$. Although this is a Bayesian estimator, you could still derive its frequentist sampling properties, including its standard error and corresponding coefficient of variation, just as you can with the …
answered Mar 16 '18 by Ben
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classical statistics) and parts that are specific to a Bayesian analysis. The statements you have made about the allowable values of the variables, and the sampling density $f(x|p)$ are model … assumptions pertaining to the sampling mechanism. They are nothing to do with the Bayesian method. Regardless of whether you are doing your analysis with Bayesian methods or classical methods, you will …
answered Mar 2 '18 by Ben
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any specification of a prior distribution. Indeed, you do not even have to be working within the Bayesian paradigm at all for these results to be applicable (see O'Neill 2009 for further discussion on …
answered Dec 30 '18 by Ben
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that one alternative is to undertake Bayesian analysis within the imprecise probability framework (see esp. Walley 1991, Walley 2000). Within this framework the prior belief is represented by a set … framework has been axiomatised by Walley as its own special form of probabilistic analysis, but is essentially equivalent to robust Bayesian analysis using a set of priors, yielding a corresponding …
answered Mar 4 by Ben
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number of sample values from the second probability vector (after the change) is affected by the inference from the first probability vector, which seems to be what you are asking about. Bayesian … Dirichlet model: Consider a hierarchical Bayesian Dirichlet model specified by: $$\mathbf{p}'|\mathbf{p} \sim \text{Di}(\kappa \cdot\boldsymbol{p}) \quad \quad \quad \quad \quad \mathbf{p} \sim \text{Di …
answered Dec 10 '18 by Ben
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Proportionality is used to simplify analysis Bayesian analysis is generally done via an even simpler statement of Bayes' theorem, where we work only in terms of proportionality with respect to the … L_\mathbf{x}(\theta) \propto \prod_{i=1}^n f(x_i|\theta).$$ This statement of Bayesian updating works in terms of proportionality with respect to the parameter $\theta$. It uses two proportionality …
answered May 31 '18 by Ben
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Let $\phi = \log (\sigma) = \tfrac{1}{2} \log (\sigma^2)$ so that you have the inverse transformation $\sigma^2 = \exp (2\phi)$. Now we apply the standard rule for transformations of random variables …
answered Jul 20 '18 by Ben
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Auto-correlation arises in the context of a stochastic time-series $\{ X_t | t \in \mathbb{Z} \}$ where we have one or more variables that vary over time. Within this context, auto-correlation is jus …
answered Jul 7 '18 by Ben
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MCMC algorithms have improving since Andrieu (2003), and we now have the NUTS sampler in Stan, which is automated so that it can be used in applied problems without the user needing to understand the …
answered Jun 28 '18 by Ben

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