```
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

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):

```
In [2]:
``````
# !wget https://raw.githubusercontent.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/master/Chapter2_MorePyMC/data/challenger_data.csv
```

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

```
In [4]:
```df['Incident'] = df['Damage Incident'].astype(float)
df

```
In [5]:
```import matplotlib.pyplot as plt
plt.scatter(df.Temperature, df.Incident, s=75, color="k",
alpha=0.5)
plt.yticks([0, 1])
plt.ylabel("Damage Incident?")
plt.xlabel("Outside temperature (Fahrenheit)")
plt.title("Defects of the Space Shuttle O-Rings vs temperature");

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

```
In [7]:
``````
# Solution goes here
```

```
In [8]:
```b0 = np.linspace(0, 50, 101);

```
In [9]:
```b1 = np.linspace(-1, 1, 101);

```
In [10]:
```from itertools import product
hypos = product(b0, b1)

```
In [11]:
```suite = Logistic(hypos);

```
In [12]:
```for data in zip(df.Temperature, df.Incident):
print(data)
suite.Update(data)

```
In [13]:
```thinkplot.Pdf(suite.Marginal(0))
thinkplot.decorate(xlabel='Intercept',
ylabel='PMF',
title='Posterior marginal distribution')

```
In [14]:
```thinkplot.Pdf(suite.Marginal(1))
thinkplot.decorate(xlabel='Log odds ratio',
ylabel='PMF',
title='Posterior marginal distribution')

```
In [15]:
``````
# Solution goes here
```

```
In [16]:
``````
# 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.

```
In [17]:
```import pymc3 as pm

```
In [23]:
``````
# Solution goes here
```

```
In [24]:
```pm.traceplot(trace);

```
In [ ]:
```