I have some data to which I'd like to fit a GLM using PyMC3. Here are the posterior predictive regression lines plotted along with the data points:
Domain knowledge indicates that a linear model with zero-intercept is appropriate and the data looks linearly related. However, it feels as though the fit is not OK since the post predictive lines seem to cover only a few data points, leaving most out. Is this intuition correct? In this case, 64% of the data points are not covered by the regression lines.
The PyMC3 model I used is really simple:
with pm.Model() as geom_glm: pm.GLM.from_formula('delta_p ~ 0 + sot', dados) step = pm.Metropolis() trace = pm.sample(draws = 10**4, step = step) plt.plot(dados['sot'], dados['delta_p'], 'x', label='dados') pm.plot_posterior_predictive_glm(trace, lm = lambda x, sample: sample['sot'] * x, eval = dados['sot'], label = 'posterior predictive check', samples = 1000) If I allow the intercept to be inferred from the data as well (even though domain knowledge indicates this is inappropriate), the situation doesn't seem to improve much:
Best Answer
It's nothing to worry about. Bayesian linear regression is (to simplify) just linear regression with random variables for the slope and intercept.
Now, the posterior distributions for the slope and intercept random variables permit many credible values for those parameters. And those combinations of credible parameters permit many credible regression lines to be drawn:
(Source for stylised image: http://www.indiana.edu/~jkkteach/WorkshopUWM2012.html)
The shaded area on your chart is simply a similar representation for the above.
I would argue the opposite to you: I would argue that you want the shaded region to be as tight as possible, to have more "confidence" (in the non-technical sense of the word) in the fitted regression line.
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