andrew_gelman_stats andrew_gelman_stats-2012 andrew_gelman_stats-2012-1247 knowledge-graph by maker-knowledge-mining
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Introduction: Konrad Scheffler writes: I was interested by your paper “Induction and deduction in Bayesian data analysis” and was wondering if you would entertain a few questions: – Under the banner of objective Bayesianism, I would posit something like this as a description of Bayesian inference: “Objective Bayesian probability is not a degree of belief (which would necessarily be subjective) but a measure of the plausibility of a hypothesis, conditional on a formally specified information state. One way of specifying a formal information state is to specify a model, which involves specifying both a prior distribution (typically for a set of unobserved variables) and a likelihood function (typically for a set of observed variables, conditioned on the values of the unobserved variables). Bayesian inference involves calculating the objective degree of plausibility of a hypothesis (typically the truth value of the hypothesis is a function of the variables mentioned above) given such a
sentIndex sentText sentNum sentScore
1 One way of specifying a formal information state is to specify a model, which involves specifying both a prior distribution (typically for a set of unobserved variables) and a likelihood function (typically for a set of observed variables, conditioned on the values of the unobserved variables). [sent-2, score-1.231]
2 Bayesian inference involves calculating the objective degree of plausibility of a hypothesis (typically the truth value of the hypothesis is a function of the variables mentioned above) given such an information state. [sent-3, score-0.96]
3 We are free to calculate probabilities conditioned on different information states and use these to argue that one information state corresponds more closely than another to a given real-world (i. [sent-4, score-0.811]
4 Alternatively we may calculate p-values conditional on an information state (via posterior predictive checking) and use them to draw conclusions about the degree to which the information state is informative about the real world. [sent-8, score-0.709]
5 ” I would not have thought this type of description should be particularly controversial, but as you point out the popular view seems to focus exclusively on subjective Bayesianism and I’m not sure where I would even find a similar description of the objective Bayesian viewpoint. [sent-12, score-0.637]
6 - Regarding your question on the continuous/discrete distinction, doesn’t it make more sense to instead distinguish between numerical and categorical variables (where the former are defined on an ordered set and the latter on an unordered set)? [sent-18, score-0.564]
7 But when a numerical variable can be replaced with a categorical variable without changing the model (i. [sent-20, score-0.367]
8 - A technical point: you claim that no prior distribution can completely reflect prior knowledge. [sent-24, score-0.474]
9 The marginal probability of data given model, p(y|M), typically depends strongly on aspects of the prior distribution that have essentially no impact on posterior inferences given the model. [sent-32, score-0.992]
10 Here’s an example of incoherence: we model some data with a normal distribution but if the rate of outliers exceeds some threshold, we switch to a t distribution. [sent-38, score-0.431]
11 The coherent thing would’ve been to start with the t distribution (if necessary, with some prior distribution that favored a large number of degrees of freedom). [sent-40, score-0.554]
12 Ultimately it comes down to setting up a reasonable joint prior distribution on the parameters at different levels of the model. [sent-46, score-0.406]
13 Scheffler responds: On item 1 above, I’m not quite sure what your point is here – the prior (which I think is best considered to be part of the model) may or may not have a strong effect on a given posterior inference. [sent-51, score-0.611]
14 This is why I think it’s important to emphasize that posterior probabilities are conditioned on the model (this seems not to be emphasized in subjective Bayesianism). [sent-52, score-0.649]
15 Regarding item 2, I agree that analysing a single data set with different distributions applied to different points is incoherent, but I haven’t seen anyone do this. [sent-55, score-0.64]
16 I don’t think it’s incoherent to use different distributions for different data sets, unless you are assuming that they are sampled from the same underlying distribution (in which case you could equally well consider them to be part of the same data set). [sent-56, score-0.788]
17 I also don’t think it’s incoherent to switch to a better model after discovering that it is better, provided you analyse the full data set with that model. [sent-57, score-0.528]
18 , that come from most statistical analyses because they depend on aspects of the model that that have essentially no impact on posterior inferences given the model. [sent-60, score-0.576]
19 Regarding item 2, my example did not involve analyzing a single data set with different distributions applied to different points. [sent-63, score-0.64]
20 I was talking about the very common procedure of using a single model for all the data points, but choosing or rejecting the model based on how it fits the data. [sent-64, score-0.572]
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