andrew_gelman_stats andrew_gelman_stats-2011 andrew_gelman_stats-2011-547 knowledge-graph by maker-knowledge-mining
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Introduction: Mike McLaughlin writes: Consider the Seeds example in vol. 1 of the BUGS examples. There, a binomial likelihood has a p parameter constructed, via logit, from two covariates. What I am wondering is: Would it be legitimate, in a binomial + logit problem like this, to allow binomial p[i] to be a function of the corresponding n[i] or would that amount to using the data in the prior? In other words, in the context of the Seeds example, is r[] the only data or is n[] data as well and therefore not permissible in a prior formulation? I [McLaughlin] currently have a model with a common beta prior for all p[i] but would like to mitigate this commonality (a kind of James-Stein effect) when there are lots of observations for some i. But this seems to feed the data back into the prior. Does it really? It also occurs to me [McLaughlin] that, perhaps, a binomial likelihood is not the one to use here (not flexible enough). My reply: Strictly speaking, “n” is data, and so what you wa
sentIndex sentText sentNum sentScore
1 Mike McLaughlin writes: Consider the Seeds example in vol. [sent-1, score-0.06]
2 There, a binomial likelihood has a p parameter constructed, via logit, from two covariates. [sent-3, score-0.747]
3 What I am wondering is: Would it be legitimate, in a binomial + logit problem like this, to allow binomial p[i] to be a function of the corresponding n[i] or would that amount to using the data in the prior? [sent-4, score-1.522]
4 In other words, in the context of the Seeds example, is r[] the only data or is n[] data as well and therefore not permissible in a prior formulation? [sent-5, score-0.538]
5 I [McLaughlin] currently have a model with a common beta prior for all p[i] but would like to mitigate this commonality (a kind of James-Stein effect) when there are lots of observations for some i. [sent-6, score-0.812]
6 But this seems to feed the data back into the prior. [sent-7, score-0.204]
7 It also occurs to me [McLaughlin] that, perhaps, a binomial likelihood is not the one to use here (not flexible enough). [sent-9, score-0.8]
8 My reply: Strictly speaking, “n” is data, and so what you want is a likelihood function p(y,n|theta), where theta represents all the parameters in the model. [sent-10, score-0.9]
9 In a binomial-type example, it would make sense to factor the likelihood as p(y|n,theta)*p(n|theta). [sent-11, score-0.324]
10 Or, to make this even clearer: p(y|n,theta_1)*p(n|theta_2), where theta_1 are the parameters of the binomial distribution (or whatever generalization you’re using) and theta_2 are the parameters involving n. [sent-12, score-0.951]
11 In any case, the next step is the prior distribution, p(theta_1,theta_2). [sent-14, score-0.247]
12 Prior dependence between theta_1 and theta_2 induces a model of the form that you’re talking about. [sent-15, score-0.445]
13 In practice, I think it can be reasonable to simplify a bit and write p(y|n,theta) and then use a prior of the form p(theta|n). [sent-16, score-0.443]
14 We discuss this sort of thing in the first or second section of the regression chapter in BDA. [sent-17, score-0.053]
15 Whether you treat n as data to be modeled or data to be conditioned on, either way you can put dependence with theta into the model. [sent-18, score-1.018]
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