andrew_gelman_stats andrew_gelman_stats-2014 andrew_gelman_stats-2014-2342 knowledge-graph by maker-knowledge-mining
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Introduction: I had an interesting conversation with Aki about monotonicity constraints. We were discussing a particular set of Gaussian processes that we were fitting to the arsenic well-switching data (the example from the logistic regression chapter in my book with Jennifer) but some more general issues arose that I thought might interest you. The idea was to fit a model where the response (the logit probability of switching wells) was constrained to be monotonically increasing in your current arsenic level and monotonically decreasing in your current distance to the closest safe well. These constraints seem reasonable enough, but when we actually fit the model we found that doing Bayesian inference with the constraint pulled the estimate, not just toward monotonicity, but to a strong increase (for the increasing relation) or a strong decrease (for the decreasing relation). This makes sense from a statistical standpoint because if you restrict a parameter to be nonnegative, any posterior dis
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1 I had an interesting conversation with Aki about monotonicity constraints. [sent-1, score-0.299]
2 We were discussing a particular set of Gaussian processes that we were fitting to the arsenic well-switching data (the example from the logistic regression chapter in my book with Jennifer) but some more general issues arose that I thought might interest you. [sent-2, score-0.67]
3 The idea was to fit a model where the response (the logit probability of switching wells) was constrained to be monotonically increasing in your current arsenic level and monotonically decreasing in your current distance to the closest safe well. [sent-3, score-2.132]
4 These constraints seem reasonable enough, but when we actually fit the model we found that doing Bayesian inference with the constraint pulled the estimate, not just toward monotonicity, but to a strong increase (for the increasing relation) or a strong decrease (for the decreasing relation). [sent-4, score-1.304]
5 This makes sense from a statistical standpoint because if you restrict a parameter to be nonnegative, any posterior distribution will end up on the positive half of the line. [sent-5, score-0.139]
6 Thinking about it more, I’m not always comfortable with any strict constraint unless there is a clear physical reason. [sent-7, score-0.499]
7 For example, yes it seems logical that increasing arsenic would increase the probability of switching but I could imagine that in any particular dataset there could be areas of negative slope. [sent-8, score-1.273]
8 After all, it is observational data and for example there could be a village that happens to have arsenic in a particular high range but where for cultural reasons there would be less switching. [sent-9, score-0.983]
9 Here there’s an omitted variable (“culture” or a village indicator) but the point is that these (hypothetical) data would not really support a strictly monotonic model, and including that restriction could distort things in other ways. [sent-10, score-0.887]
10 It does not mean we should ignore prior information, of course, but it’s a reason that I prefer soft rather than hard constraints. [sent-12, score-0.218]
11 Alternatively in this example one could put a hard constraint on the monotonicity and then add a latent omitted variable which would have the effect of turning it into a soft constraint, but I don’t usually see people do this. [sent-13, score-1.424]
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Introduction: I had an interesting conversation with Aki about monotonicity constraints. We were discussing a particular set of Gaussian processes that we were fitting to the arsenic well-switching data (the example from the logistic regression chapter in my book with Jennifer) but some more general issues arose that I thought might interest you. The idea was to fit a model where the response (the logit probability of switching wells) was constrained to be monotonically increasing in your current arsenic level and monotonically decreasing in your current distance to the closest safe well. These constraints seem reasonable enough, but when we actually fit the model we found that doing Bayesian inference with the constraint pulled the estimate, not just toward monotonicity, but to a strong increase (for the increasing relation) or a strong decrease (for the decreasing relation). This makes sense from a statistical standpoint because if you restrict a parameter to be nonnegative, any posterior dis
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Introduction: I had an interesting conversation with Aki about monotonicity constraints. We were discussing a particular set of Gaussian processes that we were fitting to the arsenic well-switching data (the example from the logistic regression chapter in my book with Jennifer) but some more general issues arose that I thought might interest you. The idea was to fit a model where the response (the logit probability of switching wells) was constrained to be monotonically increasing in your current arsenic level and monotonically decreasing in your current distance to the closest safe well. These constraints seem reasonable enough, but when we actually fit the model we found that doing Bayesian inference with the constraint pulled the estimate, not just toward monotonicity, but to a strong increase (for the increasing relation) or a strong decrease (for the decreasing relation). This makes sense from a statistical standpoint because if you restrict a parameter to be nonnegative, any posterior dis
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Introduction: I had an interesting conversation with Aki about monotonicity constraints. We were discussing a particular set of Gaussian processes that we were fitting to the arsenic well-switching data (the example from the logistic regression chapter in my book with Jennifer) but some more general issues arose that I thought might interest you. The idea was to fit a model where the response (the logit probability of switching wells) was constrained to be monotonically increasing in your current arsenic level and monotonically decreasing in your current distance to the closest safe well. These constraints seem reasonable enough, but when we actually fit the model we found that doing Bayesian inference with the constraint pulled the estimate, not just toward monotonicity, but to a strong increase (for the increasing relation) or a strong decrease (for the decreasing relation). This makes sense from a statistical standpoint because if you restrict a parameter to be nonnegative, any posterior dis
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