Think Bayes

Copyright 2018 Allen B. Downey

MIT License:

In [1]:
# Configure Jupyter so figures appear in the notebook
%matplotlib inline

# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'

import numpy as np
import pandas as pd

# import classes from thinkbayes2
from thinkbayes2 import Pmf, Cdf, Suite, Joint

import thinkplot

The Space Shuttle problem

Here's a problem from Bayesian Methods for Hackers

On January 28, 1986, the twenty-fifth flight of the U.S. space shuttle program ended in disaster when one of the rocket boosters of the Shuttle Challenger exploded shortly after lift-off, killing all seven crew members. The presidential commission on the accident concluded that it was caused by the failure of an O-ring in a field joint on the rocket booster, and that this failure was due to a faulty design that made the O-ring unacceptably sensitive to a number of factors including outside temperature. Of the previous 24 flights, data were available on failures of O-rings on 23, (one was lost at sea), and these data were discussed on the evening preceding the Challenger launch, but unfortunately only the data corresponding to the 7 flights on which there was a damage incident were considered important and these were thought to show no obvious trend. The data are shown below (see 1):

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# !wget

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columns = ['Date', 'Temperature', 'Incident']
df = pd.read_csv('challenger_data.csv', parse_dates=[0])
df.drop(labels=[3, 24], inplace=True)

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df['Incident'] = df['Damage Incident'].astype(float)

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import matplotlib.pyplot as plt

plt.scatter(df.Temperature, df.Incident, s=75, color="k",
plt.yticks([0, 1])
plt.ylabel("Damage Incident?")
plt.xlabel("Outside temperature (Fahrenheit)")
plt.title("Defects of the Space Shuttle O-Rings vs temperature");

Grid algorithm

We can solve the problem first using a grid algorithm, with parameters b0 and b1, and

$\mathrm{logit}(p) = b0 + b1 * T$

and each datum being a temperature T and a boolean outcome fail, which is true is there was damage and false otherwise.

Hint: the expit function from scipy.special computes the inverse of the logit function.

In [6]:
from scipy.special import expit

class Logistic(Suite, Joint):
    def Likelihood(self, data, hypo):
        data: T, fail
        hypo: b0, b1
        return 1

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# Solution goes here

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b0 = np.linspace(0, 50, 101);

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b1 = np.linspace(-1, 1, 101);

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from itertools import product
hypos = product(b0, b1)

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suite = Logistic(hypos);

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for data in zip(df.Temperature, df.Incident):

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                   title='Posterior marginal distribution')

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thinkplot.decorate(xlabel='Log odds ratio',
                   title='Posterior marginal distribution')

According to the posterior distribution, what was the probability of damage when the shuttle launched at 31 degF?

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# Solution goes here

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# Solution goes here


Implement this model using MCMC. As a starting place, you can use this example from the PyMC3 docs.

As a challege, try writing the model more explicitly, rather than using the GLM module.

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import pymc3 as pm

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# Solution goes here

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The posterior distributions for these parameters should be similar to what we got with the grid algorithm.

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