# Chapter 2¶

This chapter introduces more PyMC syntax and design patterns, and ways to think about how to model a system from a Bayesian perspective. It also contains tips and data visualization techniques for assessing goodness-of-fit for your Bayesian model.

## A little more on PyMC¶

### Parent and Child relationships¶

To assist with describing Bayesian relationships, and to be consistent with PyMC's documentation, we introduce parent and child variables.

• parent variables are variables that influence another variable.

• child variable are variables that are affected by other variables, i.e. are the subject of parent variables.

A variable can be both a parent and child. For example, consider the PyMC code below.

In [8]:
import pymc as mc

parameter = mc.Exponential( "poisson_param", 1 )

data_generator = mc.Poisson("data_generator", parameter )

data_plus_one = data_generator + 1


parameter controls the parameter of data_generator, hence influences its values. The former is a parent of the latter. By symmetry, data_generator is a child of parameter.

Likewise, data_generator is a parent to the variable data_plus_one (hence making data_generator both a parent and child variable). Although it does not look like one, data_plus_one should be treated as a PyMC variable as it is a function of another PyMC variable, hence is a child variable to data_generator.

This nomenclature is introduced to help us describe relationships in PyMC modeling. You can access a variables children and parent variables using the children and parents attributes attached to variables.

In [9]:
print "Children of parameter: "
print parameter.children
print "\nParents of data_generator: "
print data_generator.parents
print "\nChildren of data_generator: "
print  data_generator.children

Children of parameter:
set([<pymc.distributions.Poisson 'data_generator' at 0x0000000008DF5198>])

Parents of data_generator:
{'mu': <pymc.distributions.Exponential 'poisson_param' at 0x0000000008DF57B8>}

Children of data_generator:


Of course a child can have more than one parent, and a parent can have many children.

### PyMC Variables¶

All PyMC variables also expose a value attribute. This method produces the current (possibly random) internal value of the variable. If the variable is a child variable, its value changes given the variable's parents' values. Using the same variables from before:

In [10]:
print "parameter.value =",parameter.value
print "data_generator.value =",data_generator.value
print "data_plus_one.value =", data_plus_one.value

parameter.value = 2.13659208293
data_generator.value = 3
data_plus_one.value = 4


PyMC is concerned with two types of programming variables: stochastic and deterministic.

• stochastic variables are variables that are not deterministic, i.e., even if you knew all the values of the variables' parents (if it even has any parents), it would still be random. Included in this category are instances of classes Poisson, DiscreteUniform, and Exponential.

• deterministic variables are variables that are not random if the variables' parents were known. This might be confusing at first: a quick mental check is if I knew all of variable foo's parent variables, I could determine what foo's value is.

We will detail each below.

#### Initializing Stochastic variables¶

Initializing a stochastic variable requires a name argument, plus additional parameters that are class specific. For example:

some_variable = mc.DiscreteUniform( "discrete_uni_var", 0, 4 )

where 0,4 are the DiscreteUniform-specific upper and lower bound on the random variable. The PyMC docs contain the specific parameters for stochastic variables. (Or use ?? if you are using IPython!)

The name attribute is used to retrieve the posterior distribution later in the analysis, so it is best to use a descriptive name. Typically, I use the Python variable's name as the name.

For multivariable problems, rather than creating a Python array of stochastic variables, addressing the size keyword in the call to a Stochastic variable creates multivariate array of (independent) stochastic variables. The array behaves like a Numpy array when used like one, and references to its value attribute return Numpy arrays.

The size argument also solves the annoying case where you may have many variables $\beta_i, \; i = 1,...,N$ you wish to model. Instead of creating arbitrary names and variables for each one, like:

beta_1 = mc.Uniform( "beta_1", 0, 1)
beta_2 = mc.Uniform( "beta_2", 0, 1)
...



we can instead wrap them into a single variable:

betas = mc.Uniform( "betas", 0, 1, size = N )



#### Calling random()¶

We can also call on a stochastic variable's random() method, which (given the parent values) will generate a new, random value. Below we demonstrate this using the texting example from the previous chapter.

In [11]:
lambda_1 = mc.Exponential( "lambda_1",  1 ) #prior on first behaviour
lambda_2 = mc.Exponential( "lambda_2", 1 ) #prior on second behaviour
tau = mc.DiscreteUniform( "tau", lower = 0, upper = 10 ) #prior on behaviour change

print "lambda_1.value = %.3f"%lambda_1.value
print "lambda_2.value = %.3f"%lambda_2.value
print "tau.value = %.3f"%tau.value
print

lambda_1.random(), lambda_2.random(), tau.random()

print "After calling random() on the variables..."
print "lambda_1.value = %.3f"%lambda_1.value
print "lambda_2.value = %.3f"%lambda_2.value
print "tau.value = %.3f"%tau.value

lambda_1.value = 0.450
lambda_2.value = 0.807
tau.value = 6.000

After calling random() on the variables...
lambda_1.value = 0.847
lambda_2.value = 3.247
tau.value = 3.000


The call to random stores a new value into the variable's value attribute. In fact, this new value is stored in the computer's cache for faster recall and efficiency.

### Warning: Don't update stochastic variables' values in-place.¶

Straight from the PyMC docs, we quote [4]:

Stochastic objects' values should not be updated in-place. This confuses PyMC's caching scheme... The only way a stochastic variable's value should be updated is using statements of the following form:

    A.value = new_value



The following are in-place updates and should never be used:

    A.value += 3
A.value[2,1] = 5
A.value.attribute = new_attribute_value

#### Deterministic variables¶

Since most variables you will be modeling are stochastic, we distinguish deterministic variables with a pymc.deterministic wrapper. (If you are unfamiliar with Python wrappers (also called decorators), that's no problem. Just prepend the pymc.deterministic decorator before the variable declaration and you're good to go. No need to know more. ) The declaration of a deterministic variable uses a Python function:

@mc.deterministic
def some_deterministic_var(v1=v1,):
#jelly goes here.



For all purposes, we can treat the object some_deterministic_var as a variable and not a Python function.

Prepending with the wrapper is the easiest way, but not the only way, to create deterministic variables. This is not completely true: elementary operations, like addition, exponentials etc. implicitly create deterministic variables. For example, the following returns a deterministic variable:

In [12]:
type( lambda_1 + lambda_2 )

Out[12]:
pymc.PyMCObjects.Deterministic

The use of the deterministic wrapper was seen in the previous chapter's text-message example. Recall the model for $\lambda$ looked like:

$$\lambda = \cases{ \lambda_1 & \text{if } t \lt \tau \cr \lambda_2 & \text{if } t \ge \tau }$$

And in PyMC code:

In [13]:
n_data_points = 5 # in CH1 we had ~70 data points

@mc.deterministic
def lambda_( tau = tau, lambda_1 = lambda_1, lambda_2 = lambda_2 ):
out = np.zeros(n_data_points)
out[:tau] = lambda_1 #lambda before tau is lambda1
out[tau:] = lambda_2 #lambda after tau is lambda1
return out


Clearly, if $\tau, \lambda_1$ and $\lambda_2$ are known, then $\lambda$ is known completely, hence it is a deterministic variable.

Inside the deterministic decorator, the Stochastic variables passed in behave like scalars or Numpy arrays ( if multivariable), and not like Stochastic variables. For example, running the following:

@mc.deterministic
def some_deterministic( stoch = some_stochastic_var ):
return stoch.value**2



will return an AttributeError detailing that stoch does not have a value attribute. It simply needs to be stoch**2. During the learning phase, it the variables value that is repeatedly passed in, not the actual variable.

Notice in the creation of the deterministic function we added defaults to each variable used in the function. This is a necessary step, and all variables must have default values.

### Including observations in the Model¶

At this point, it may not look like it, but we have fully specified our priors. For example, we can ask and answer questions like "What does my prior distribution of $\lambda_1$ look like?"

In [14]:
%pylab inline
figsize(12.5, 4)

samples = [ lambda_1.random() for i in range( 20000) ]
hist( samples, bins = 70, normed=True )
plt.title( "Prior distribution for $\lambda_1$")
plt.xlim( 0, 8);

Out[14]:
(0, 8)

To frame this in the notation of the first chapter, though this is a slight abuse of notation, we have specified $P(A)$. Our next goal is to include data/evidence/observations $X$ into our model.

PyMC stochastic variables have a keyword argument observed which accepts a boolean (False by default). The keyword observed has a very simple role: fix the variable's current value, i.e. make value immutable. We have to specify an initial value in the variable's creation, equal to the observations we wish to include, typically an array (and it should be an Numpy array for speed). For example:

In [17]:
data = np.array( [10, 5] )
fixed_variable = mc.Poisson( "fxd", 1, value = data, observed = True )
print "value: ",fixed_variable.value
print "calling .random()"
fixed_variable.random()
print "value: ",fixed_variable.value

value:  [10  5]
calling .random()
value:  [10  5]


This is how we include data into our models: initializing a stochastic variable to have a fixed value.

To complete our text message example, we fix the PyMC variable observations to the observed dataset.

In [18]:
#we're using some fake data here
data = np.array( [ 10, 25, 15, 20, 35] )
obs = mc.Poisson( "obs", lambda_, value = data, observed = True )
print obs.value

[10 25 15 20 35]


### Finally...¶

We wrap all the created variables into a mc.Model class. With this Model class, we can analyze the variables as a single unit.

In [19]:
model = mc.Model( [obs, lambda_, lambda_1, lambda_2, tau] )


## Modeling approaches¶

A good starting thought to Bayesian modeling is to think about how your data might have been generated. Position yourself in an omniscient position, and try to imagine how you would recreate the dataset.

In the last chapter we investigated text message data. We begin by asking how our observations may have been generated:

1. We started by thinking "what is the best random variable to describe this count data?" A Poisson random variable is a good candidate because it can represent count data. So we model the number of sms's received as sampled from a Poisson distribution.

2. Next, we think, "Ok, assuming sms's are Poisson-distributed, what do I need for the Poisson distribution?" Well, the Poisson distribution has a parameters $\lambda$.

3. Do we know $\lambda$? No. In fact, we have a suspicion that there are two $\lambda$ values, one for the earlier behaviour and one for the latter behaviour. We don't know when the behaviour switches though, but call the switchpoint $\tau$.

4. What is a good distribution for the two $\lambda$s? The exponential is good, as it assigns probabilities to positive real numbers. Well the exponential distribution has a parameter too, call it $\alpha$.

5. Do we know what the parameter $\alpha$ might be? No. At this point, we could continue and assign a distribution to $\alpha$, but it's better to stop once we reach a set level of ignorance: whereas we have a prior belief about $\lambda$, ("it probably changes over time", "it's likely between 10 and 30", etc.), we don't really have any strong beliefs about $\alpha$. So it's best to stop here.

What is a good value for $\alpha$ then? We think that the $\lambda$s are between 10-30, so if we set $\alpha$ really low (which corresponds to larger probability on high values) we are not reflecting our prior well. Similar, a too-high alpha misses our prior belief as well. A good idea for $\alpha$ as to reflect our belief is to set the value so that the mean of $\lambda$, given $\alpha$, is equal to our observed mean. This was shown in the last chapter.

6. We have no expert opinion of when $\tau$ might have occurred. So we will suppose $\tau$ is from a discrete uniform distribution over the entire timespan.

Below we give a graphical visualization of this, where arrows denote parent-child relationships. (provided by the Daft Python library )

<img src="http://i.imgur.com/7J30oCG.png" width = 700/>

PyMC, and other probabilistic programming languages, have been designed to tell these data-generation stories. More generally, B. Cronin writes [5]:

Probabilistic programming will unlock narrative explanations of data, one of the holy grails of business analytics and the unsung hero of scientific persuasion. People think in terms of stories - thus the unreasonable power of the anecdote to drive decision-making, well-founded or not. But existing analytics largely fails to provide this kind of story; instead, numbers seemingly appear out of thin air, with little of the causal context that humans prefer when weighing their options.

### Same story; different ending.¶

Interestingly, we can create new datasets by retelling the story. For example, if we reverse the above steps, we can simulate a possible realization of the dataset.

1. Specify when the user's behaviour switches by sampling from $\text{DiscreteUniform}(0, 80)$:

In [20]:
tau = mc.rdiscrete_uniform(0, 80)
print tau

63


2. Draw $\lambda_1$ and $\lambda_2$ from an $\text{Exp}(\alpha)$ distribution:

In [21]:
alpha = 1./20.
lambda_1, lambda_2 = mc.rexponential( alpha, 2 )
print lambda_1, lambda_2

16.7630558157 33.1568046968


3. For days before $\tau$, represent the user's received SMS count by sampling from $\text{Poi}(\lambda_1)$, and sample from $\text{Poi}(\lambda_2)$ for days after $\tau$. For example:

In [22]:
data = np.r_[ mc.rpoisson( lambda_1, tau ), mc.rpoisson( lambda_2,80 - tau) ]


4. Plot the artificial dataset:

In [23]:
plt.bar( np.arange( 80 ), data, color ="#348ABD"  )
plt.bar( tau-1, data[tau-1], color = "r", label = "user behaviour changed" )
plt.xlabel( "Time (days)")
plt.title("Artifical dataset")
plt.xlim( 0, 80 );
plt.legend();

Out[23]:
<matplotlib.legend.Legend at 0x12f67710>

It is okay that our fictional dataset does not look like our observed dataset: the probability is incredibly small it indeed would. PyMC's engine is designed to find good parameters, $\lambda_i, \tau$, that maximize this probability.

The ability to generate artifical dataset is an interesting side effect of our modeling, and we will see that this ability is a very important method of Bayesian inference. We produce a few more datasets below:

In [29]:
def plot_artificial_sms_dataset():
tau = mc.rdiscrete_uniform(0, 80)
alpha = 1./20.
lambda_1, lambda_2 = mc.rexponential( alpha, 2 )
data = np.r_[ mc.rpoisson( lambda_1, tau ), mc.rpoisson( lambda_2,80- tau) ]
plt.bar( np.arange( 80 ), data, color ="#348ABD"  )
plt.bar( tau-1, data[tau-1], color = "r", label = "user behaviour changed" )
plt.xlim( 0, 80 );

figsize( 12.5,5)
plt.title("More example of artificial datasets" )
for i in range(4):
plt.subplot(4,1,i)
plot_artificial_sms_dataset()


Later we will see how we use this to make predictions and test the appropriateness of our models.

## An algorithm for human deceit¶

Likely the most common statistical task is estimating the frequency of events. However, there is a difference between the observed frequency and the true frequency of an event. The true frequency can be interpreted as the probability of an event occurring. For example, the true frequency of rolling a 1 on a 6-sided die is 0.166. Knowing the frequency of events like baseball home runs, frequency of social attributes, fraction of internet users with cats etc. are common requests we ask of Nature. Unfortunately, in general Nature hides the true frequency from us and we must infer it from observed data.

The observed frequency is then the frequency we observe: say rolling the die 100 times you may observe 20 rolls of 1. The observed frequency, 0.2, differs from the true frequency, 0.166. We can use Bayesian statistics to infer probable values of the true frequency using an appropriate prior and observed data.

Social data is really interesting as people are not always honest with responses, which adds a further complication into inference. For example, simply asking individuals "Have you ever cheated on a test?" will surely contain some rate of dishonesty. What you can say for certain is that the true rate is less than your observed rate (assuming individuals lie only about not cheating; I cannot imagine one who would admit "Yes" to cheating when in fact they hadn't cheated).

To present an elegant solution to circumventing this dishonesty problem, and to demonstrate Bayesian modeling, we first need to introduce the binomial distribution.

### The Binomial Distribution¶

The binomial distribution is one of the most popular distributions, mostly because of its simplicity and usefulness. Unlike the other distributions we have encountered thus far in the book, the binomial distribution has 2 parameters: $N$, a positive integer representing $N$ trials or number of instances of potential events, and $p$, the probability of an event occurring in a single trial. Like the Poisson distribution, it is a discrete distribution, but unlike the Poisson distribution, it only weighs integers from $0$ to $N$. The mass distribution looks like:

$$P( X = k ) = {{N}\choose{k}} p^k(1-p)^{N-k}$$

If $X$ is a binomial random variable with parameters $p$ and $N$, denoted $X \sim \text{Bin}(N,p)$, then $X$ is the number of events that occured in the $N$ trials (obviously $0 \le X \le N$). The larger $p$ is (while still remaining between 0 and 1), the more events are likely to occur. The expected value of a binomial is equal to $Np$. Below we plot the mass probability distribution for varying parameters.

In [22]:
figsize( 12.5, 4)

import scipy.stats as stats
binomial = stats.binom

parameters = [ (10, .4) , (10, .9) ]
colors = ["#348ABD", "#A60628"]

for i in range(2):
N, p = parameters[i]
_x = np.arange( N+1 )
plt.bar( _x - 0.5,  binomial.pmf( _x, N, p ), color = colors[i],
edgecolor=colors[i],
alpha = 0.6,
label = "$N$: %d, $p$: %.1f"%(N,p),
linewidth=3)

plt.legend(loc="upper left")
plt.xlim(0, 10.5)
plt.xlabel("$k$")
plt.ylabel("$P(X = k)$")
plt.title("Probability mass distributions of binomial random variables");


The special case when $N = 1$ corresponds to the Bernoulli distribution. If $X\ \sim \text{Ber}(p)$, then $X$ is 1 with probability $p$ and 0 with probability $1-p$. The Bernoulli distribution is useful for indicators, e.g. $Y = X\alpha + (1-X)\beta$ is $\alpha$ with probability $p$ and $\beta$ with probability $1-p$.

There is another connection between Bernoulli and Binomial random variables. If we have $X_1, X_2, ... , X_N$ Bernoulli random variables with the same $p$, then $Z = X_1 + X_2 + ... + X_N \sim \text{Binomial}(N, p )$.

The expected value of a Bernoulli random variable is $p$. This can be seen by noting the more general Binomial random variable has expected value $Np$ and setting $N=1$.

##### Example: Cheating among students¶

We will use the binomial distribution to determine the frequency of students cheating during an exam. If we let $N$ be the total number of students who took the exam, and assuming each student is interviewed post-exam (answering without consequence), we will receive integer $X$ "Yes I did cheat" answers. We then find the posterior distribution of $p$, given $N$, some specified prior on $p$, and observed data $X$.

This is a completely absurd model. No student, even with a free-pass against punishment, would admit to cheating. What we need is a better algorithm to ask students if they had cheated. Ideally the algorithm should encourage individuals to be honest while preserving privacy. The following proposed algorithm is a solution I greatly admire for its ingenuity and effectiveness:

In the interview process for each student, the student flips a coin, hidden from the interviewer. The student agrees to answer honestly if the coin comes up heads. Otherwise, if the coin comes up tails, the student (secretly) flips the coin again, and answers "Yes, I did cheat" if the coin flip lands heads, and "No, I did not cheat", if the coin flip lands tails. This way, the interviewer does not know if a "Yes" was the result of a guilty plea, or a Heads on a second coin toss. Thus privacy is preserved and the researchers receive honest answers.

I call this the Privacy Algorithm. One could of course argue that the interviewers are still receiving false data since some "Yes"'s are not confessions but instead randomness, but an alternative perspective is that the researchers are discarding approximately half of their original dataset since half of the responses will be noise. But they have gained a systematic data generation process that can be modeled. Furthermore, they do not have to incorporate (perhaps somewhat naively) the possibility of deceitful answers. We can use PyMC to dig through this noisy model, and find a posterior distribution for the true frequency of liars.

Suppose 100 students are being surveyed for cheating, and we wish to find $p$, the proportion of cheaters. There a few ways we can model this in PyMC. I'll demonstrate the most explicit way, and later show a simplified version. Both versions arrive at the same inference. In our data-generation model, we sample $p$, the true proportion of cheaters, from a prior. Since we are quite ignorant about $p$, we will assign it a $\text{Uniform}(0,1)$ prior.

In [3]:
import pymc as mc

N = 100

p = mc.Uniform( "freq_cheating", 0, 1)


Again, thinking of our data-generation model, we assign Bernoulli random variables to the 100 students: 1 implies they cheated and 0 implies they did not.

In [4]:
true_answers = mc.Bernoulli( "truths", p, size = N)


If we carry out the algorithm, the next step that occurs is the first coin-flip each student makes. This can be modeled again by sampling 100 Bernoulli random variables with $p=1/2$: denote a 1 as a Heads and 0 a Tails.

In [5]:
first_coin_flips = mc.Bernoulli( "first_flips", 0.5, size = N)
print first_coin_flips.value

[False  True  True False  True  True False  True  True  True  True  True
False  True  True  True False  True False  True  True False False  True
True False  True False  True False False False False False  True False
True False  True False False  True  True False  True False False False
False False  True False False  True  True  True  True  True False False
False  True  True False  True False  True False  True  True  True  True
False  True False  True False  True  True False  True  True  True False
False  True  True False False  True  True  True False False False  True
True  True False False]


Although not everyone flips a second time, we can still model the possible realization of second coin-flips:

In [6]:
second_coin_flips = mc.Bernoulli("second_flips", 0.5, size = N)


Using these variables, we can return a possible realization of the observed proportion of "Yes" responses. We do this using a PyMC deterministic variable:

In [7]:
@mc.deterministic
fc = first_coin_flips,
sc = second_coin_flips ):

observed = fc*t_a + (1-fc)*sc
return observed.sum()/float(N)


The line fc*t_a + (1-fc)*sc contains the heart of the Privacy algorithm. Elements in this array are 1 if and only if i) the first toss is heads and the student cheated or ii) the first toss is tails, and the second is heads, and are 0 else. Summing this vector and dividing by float(N) produces a proportion.

In [8]:
observed_proportion.value

Out[8]:
0.32000000000000001

Next we need a dataset. After performing our coin-flipped interviews the researchers received 35 "Yes" responses. To put this into a relative perspective, if there truly were no cheaters, we should expect to see on average 1/4 of all responses being a "Yes" (half chance of having first coin land Tails, and another half chance of having second coin land Heads), so about 25 responses in a cheat-free world. On the other hand, if all students cheated, we should expected to see on approximately 3/4 of all response be "Yes".

The researchers observe a Binomial random variable, with N = 100 and p = observed_proportion with value = 35:

In [9]:
X = 35

observations = mc.Binomial("obs", N, observed_proportion, observed = True, value = X)


Below we add all the variables of interest to a Model container and run our black-box algorithm over the model.

In [14]:
model = mc.Model( [p, true_answers, first_coin_flips,
second_coin_flips, observed_proportion, observations] )

### To be explained in Chapter 3!
mcmc = mc.MCMC( model )
mcmc.sample( 150000, 120000,4 )

[****************100%******************]  150000 of 150000 complete

In [15]:
figsize(12.5, 3 )
from scipy.stats.mstats import mquantiles

p_trace = mcmc.trace("freq_cheating")[:]
quantiles =mquantiles( p_trace, prob=[0.05, 0.95] )

plt.title("Posterior distribution of frequency of cheaters")
plt.hist( p_trace, histtype="stepfilled" , normed = True,
alpha = 0.85, bins = 30, label = "posterior distribution",
color = "#348ABD")
plt.vlines( quantiles, [0,0], [5,5], linestyles = "--" )
plt.xlim(0,1)
plt.legend();


With regards to the above plot, we are still pretty uncertain about what the true frequency of cheaters might be, but we have narrowed the true frequency down to a range between 0.05 to 0.45 (marked by the dashed lines). This is pretty good, as a priori we had no idea how many students might have cheated (hence the uniform distribution for our prior). On the other hand, it is also pretty bad since there is a .4 length window the true value most likely lives in. Have we even gained anything, or are we still too uncertain about the true frequency?

I would argue, yes, we have discovered something. It is implausible, according to our posterior, that there are no cheaters, i.e. the posterior assigns low probability to $p=0$. Since we started with an uniform prior, treating all values of $p$ as equally plausible, but the data ruled out $p=0$ as a possibility, we can be confident that there were cheaters.

This kind of algorithm can be used to gather private information from users and be reasonably confident that the data, though noisy, is truthful.

### Alternative PyMC Model¶

Given a value for $p$ (which from our god-like position we know), we can find the probability the student will answer yes:

\begin{align} P(\text{"Yes"}) = & P( \text{Heads on first coin} )P( \text{cheater} ) + P( \text{Tails on first coin} )P( \text{Heads on second coin} ) \\\\ & = \frac{1}{2}p + \frac{1}{2}\frac{1}{2}\\\\ & = \frac{p}{2} + \frac{1}{4} \end{align}

Thus, knowing $p$ we know the probability a student will respond "Yes". In PyMC, we can create a deterministic function to evaluate the probability of responding "Yes", given $p$:

In [17]:
p = mc.Uniform( "freq_cheating", 0, 1)

@mc.deterministic
def p_skewed( p =  p ):
return 0.5*p + 0.25


I could have typed p_skewed = 0.5*p + 0.25 instead for a one-liner, as the elementary operations of addition and scalar multiplication will implicitly create a deterministic variable, but I wanted to make the deterministic boilerplate explicit for clarity's sake.

If we know the probability of respondents saying "Yes", which is p_skewed, and we have $N=100$ students, the number of "Yes" responses is a binomial random variable with parameters N and p_skewed.

This is were we include our observed 35 "Yes" responses. In the declaration of the mc.Binomial, we include value = 35 and observed = True.

In [18]:
yes_responses = mc.Binomial( "number_cheaters", 100, p_skewed,
value = 35, observed = True )


Below we add all the variables of interest to a Model container and run our black-box algorithm over the model.

In [19]:
model = mc.Model( [yes_responses, p_skewed, p ] )

### To Be Explained in Chapter 3!
mcmc = mc.MCMC(model)
mcmc.sample( 12500, 2500 )

[****************100%******************]  12500 of 12500 complete

In [20]:
figsize(12.5, 3 )
p_trace = mcmc.trace("freq_cheating")[:]
quantiles =mquantiles( p_trace, prob=[0.05, 0.95] )

plt.title("Posterior distribution of frequency of cheaters")
plt.hist( p_trace, histtype="stepfilled" , normed = True,
alpha = 0.85, bins = 30, label = "posterior distribution",
color = "#348ABD")
plt.vlines( quantiles, [0,0], [5,5], linestyles = "--" )
plt.xlim(0,1)
plt.legend();


### More PyMC Tricks¶

#### Protip: Lighter deterministic variables with Lambda class¶

Sometimes writing a deterministic function using the @mc.deterministic decorator can seem like a chore, especially for a small function. I have already mentioned that elementary math operations can produce deterministic variables implicitly, but what about operations like indexing or slicing? Built-in Lambda functions can handle this with the elegance and simplicity required. For example,

beta = mc.Normal( "coefficients", 0, size=(N,1) )
x = np.random.randn( (N,1) )
linear_combination = mc.Lambda( lambda x=x, beta = beta: np.dot( x.T, beta ) )



#### Protip: Arrays of PyMC variables¶

There is no reason why we cannot store multiple heterogeneous PyMC variables in a Numpy array. Just remember to set the dtype of the array to object upon initialization. For example:

In [16]:
N = 10
x = np.empty( N , dtype=object )
for i in range(0, N):
x[i] = mc.Exponential('x_%i' % i, (i+1)**2)


The remainder of this chapter examines some practical examples of PyMC and PyMC modeling:

##### Example: Challenger Space Shuttle Disaster ¶

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 [24]:
figsize( 12.5, 3.5 )
np.set_printoptions(precision=3, suppress= True)
challenger_data = np.genfromtxt("data/challenger_data.csv", skip_header = 1, \
usecols=[1,2], missing_values="NA", delimiter=",")
#drop the NA values
challenger_data = challenger_data[ ~np.isnan(challenger_data[:,1]) ]

#plot it, as a function of tempature (the first column)
print "Temp (F), O-Ring failure?"
print challenger_data

plt.scatter( challenger_data[:,0], challenger_data[:,1], 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");

Temp (F), O-Ring failure?
[[ 66.   0.]
[ 70.   1.]
[ 69.   0.]
[ 68.   0.]
[ 67.   0.]
[ 72.   0.]
[ 73.   0.]
[ 70.   0.]
[ 57.   1.]
[ 63.   1.]
[ 70.   1.]
[ 78.   0.]
[ 67.   0.]
[ 53.   1.]
[ 67.   0.]
[ 75.   0.]
[ 70.   0.]
[ 81.   0.]
[ 76.   0.]
[ 79.   0.]
[ 75.   1.]
[ 76.   0.]
[ 58.   1.]]


It looks clear that the probability of damage incidents occurring increases as the outside temperature decreases. We are interested in modeling the probability here because it does not look like there is a strict cutoff point between temperature and a damage incident occurring. The best we can do is ask "At temperature $t$, what is the probability of a damage incident?". The goal of this example is to answer that question.

We need a function of temperature, call it $p(t)$, that is bounded between 0 and 1 (so as to model a probability) and changes from 1 to 0 as we increase temperature. There are actually many such functions, but the most popular choice is the logistic function.

$$p(t) = \frac{1}{ 1 + e^{ \;\beta t } }$$

In this model, $\beta$ is the variable we are uncertain about. Below is the function plotted for $\beta = 1, 3, -5$.

In [25]:
figsize(12,3)

def logistic( x, beta):
return 1.0/( 1.0 + np.exp( beta*x) )

x = np.linspace( -4, 4, 100 )
plt.plot(x, logistic( x, 1), label = r"$\beta = 1$")
plt.plot(x, logistic( x, 3), label = r"$\beta = 3$")
plt.plot(x, logistic( x, -5), label = r"$\beta = -5$")
plt.legend();


But something is missing. In the plot of the logistic function, the probability changes only near zero, but in our data above the probability changes around 65 to 70. We need to add a bias term to our logistic function:

$$p(t) = \frac{1}{ 1 + e^{ \;\beta t + \alpha } }$$

Some plots are below, with differing $\alpha$.

In [27]:
def logistic( x, beta, alpha=0):
return 1.0/( 1.0 + np.exp( np.dot(beta, x) + alpha) )

x = np.linspace( -4, 4, 100 )

plt.plot(x, logistic( x, 1), label = r"$\beta = 1$", ls= "--", lw =1 )
plt.plot(x, logistic( x, 3), label = r"$\beta = 3$", ls= "--", lw =1)
plt.plot(x, logistic( x, -5), label = r"$\beta = -5$", ls= "--", lw =1)

plt.plot(x, logistic( x, 1,1), label = r"$\beta = 1, \alpha = 1$",
color = "#348ABD")
plt.plot(x, logistic( x, 3, -2), label = r"$\beta = 3, \alpha = -2$",
color="#A60628")
plt.plot(x, logistic( x, -5, 7), label = r"$\beta = -5, \alpha = 7$",
color = "#7A68A6")

plt.legend(loc = "lower left");


Adding a constant term $\alpha$ amounts to shifting the curve left or right (hence why it is called a bias. )

Let's start modeling this in PyMC. The $\beta, \alpha$ paramters have no reason to be positive, bounded or relatively large, so they are best modeled by a Normal random variable, introduced next.

### Normal distributions¶

A Normal random variable, denoted $X \sim N(\mu, 1/\tau)$, has a distribution with two parameters: the mean, $\mu$, and the precision, $\tau$. Those familar with the Normal distribution already have probably seen $\sigma^2$ instead of $\tau$. They are in fact reciprocals of each other. The change was motivated by simpler mathematical analysis and is an artifact of older Bayesian methods. Just remember: The smaller $\tau$, the larger the spread of the distribution (i.e. we are more uncertain); the larger $\tau$, the tighter the distribution (i.e. we are more certain). Regardless, $\tau$ is always positive.

The probability density function of a $N( \mu, 1/\tau)$ random variable is:

$$f(x | \mu, \tau) = \sqrt{\frac{\tau}{2\pi}} \exp\left( -\frac{\tau}{2} (x-\mu)^2 \right)$$

We plot some different density functions below.

In [28]:
import scipy.stats as stats

nor = stats.norm
x = np.linspace( -8, 7, 150 )
mu = (-2, 0, 3)
tau = (.7, 1, 2.8 )
colors = ["#348ABD", "#A60628", "#7A68A6"]
parameters = zip( mu, tau, colors )

for _mu, _tau, _color in parameters:
plt.plot( x, nor.pdf( x, _mu , scale = 1./_tau ), \
label ="$\mu = %d,\;\\tau = %.1f$"%(_mu, _tau), color = _color )
plt.fill_between( x, nor.pdf( x, _mu, scale =1./_tau ), color = _color, \
alpha = .33)

plt.legend(loc = "upper right")
plt.xlabel("$x$")
plt.ylabel("density function at $x$")
plt.title("Probability distribution of three different Normal random variables");


A Normal random variable can be take on any real number, but the variable is very likely to be relatively close to $\mu$. In fact, the expected value of a Normal is equal to its $\mu$ parameter:

$$E[ X | \mu, \tau] = \mu$$

and its variance is equal to the inverse of $\tau$:

$$Var( X | \mu, \tau ) = \frac{1}{\tau}$$

Below we continue our modeling of the Challenger space craft:

In [29]:
import pymc as mc

temperature = challenger_data[:,0]
D = challenger_data[:,1] #defect or not?

#notice thevalue here. We explain why below.
beta = mc.Normal( "beta", 0, 0.001, value = 0 )
alpha = mc.Normal( "alpha", 0, 0.001, value = 0 )

@mc.deterministic
def p( t = temperature, alpha = alpha, beta = beta):
return 1.0/( 1. + np.exp( beta*t + alpha) )


We have our probabilities, but how do we connect them to our observed data? A Bernoulli random variable with parameter $p$, denoted $\text{Ber}(p)$, is a random variable that takes value 1 with probability $p$, and 0 else. Thus, our model can look like:

$$\text{Defect Incident, D_i} \sim \text{Ber}( \;p(t_i)\; ), \;\; i=1..N$$

where $p(t)$ is our logistic function and $t_i$ are the temperatures we have observations about. Notice in the above code we had to set the values of beta and alpha to 0. The reason for this is that if beta and alpha are very large, they make p equal to 1 or 0. Unfortunately, mc.Bernoulli does not like probabilities of exactly 0 or 1, though they are mathematically well-defined probabilities. So by setting the coefficient values to 0, we set the variable p to be a reasonable starting value. This has no effect on our results, nor does it mean we are including any additional information in our prior. It is simply a computational caveat in PyMC.

In [30]:
p.value

Out[30]:
array([ 0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,  0.5,
0.5])
In [48]:
# connect the probabilities in p with our observations through a
# Bernoulli random variable.
observed = mc.Bernoulli( "bernoulli_obs", p, value = D, observed=True)

model = mc.Model( [observed, beta, alpha] )

#mysterious code to be explained in Chapter 3
map_ = mc.MAP(model)
map_.fit()
mcmc = mc.MCMC( model )
mcmc.sample( 120000, 100000, 2 )

[****************100%******************]  120000 of 120000 complete


We have trained our model on the observed data, now we can sample values from the posterior. Let's look at the posterior distributions for $\alpha$ and $\beta$:

In [51]:
alpha_samples = mcmc.trace( 'alpha' )[:, None] #best to make them 1d
beta_samples = mcmc.trace( 'beta' )[:, None]

figsize(12.5, 6)

#histogram of the samples:
plt.subplot(211)
plt.title(r"Posterior distributions of the variables $\alpha, \beta$")
plt.hist( beta_samples, histtype='stepfilled', bins = 35, alpha = 0.85, \
label = r"posterior of $\beta$", color = "#7A68A6",normed = True )
plt.legend()

plt.subplot(212)
plt.hist( alpha_samples, histtype='stepfilled', bins = 35, alpha = 0.85, \
label = r"posterior of $\alpha$", color = "#A60628",normed = True )
plt.legend();


Notice that neither distribution has postive probability at 0. What would it mean if a variable assigned positive probability at 0? If the coefficient was zero, then it would not influence the probability of defect. The larger the probability at 0, the larger the probability the variable has no effect on the outcome. In this example though, both variables are significant to the outcome.

Regarding the spread of the data, we are very uncertain about what the true parameters might be (though considering the low sample size and the large overlap of defects-to-nondefects this behaviour is perhaps expected).

Next, let's look at the expected probability for a specific value of the temperature. That is, we average over all samples from the posterior to get a likely value for $p(t_i)$.

In [52]:
t = np.linspace( temperature.min() - 5, temperature.max()+5, 50 )[:,None]
p_t= logistic( t.T, beta_samples, alpha_samples )

mean_prob_t = p_t.mean(axis=0)

In [54]:
figsize( 12.5, 4)

plt.plot( t, mean_prob_t, lw = 3, label = "average posterior \nprobability \
of defect")
plt.plot( t, p_t[0, :], ls="--",label="realization from posterior" )
plt.plot( t, p_t[-2, :], ls="--", label="realization from posterior"  )
plt.scatter( temperature, D, color = "k", s = 50, alpha = 0.5 )
plt.title("Posterior expected value of probability of defect; plus realizations")
plt.legend(loc= "lower left")
plt.ylim( -0.1, 1.1 )
plt.xlim( t.min(), t.max() )
plt.ylabel("probability")
plt.xlabel("temperature");


Above we also plotted two possible realizations of what the actual underlying system might be. Both are equally likely as any other draw. The blue line is what occurs when we average all the 20000 possible dotted lines together.

An interesting question to ask is for what temperatures are we most uncertain about the defect-probability? Below we plot the expected value line and the associated 95% intervals for each temperature.

In [55]:
from scipy.stats.mstats import mquantiles

# vectorized bottom and top 5% quantiles for "confidence interval"
qs = mquantiles(p_t,[0.05,0.95],axis=0)
plt.fill_between(t[:,0],*qs,alpha = 0.7,
color = "#7A68A6")

plt.plot(t[:,0], qs[0], label="95% CI", color = "#7A68A6", alpha =0.7)

plt.plot( t, mean_prob_t, lw = 1, ls= "--", color = "k",
label = "average posterior \nprobability of defect")

plt.xlim( t.min(), t.max() )
plt.ylim( -0.02, 1.02 )
plt.legend(loc="lower left")
plt.scatter( temperature, D, color = "k", s = 50, alpha = 0.5 )
plt.xlabel("temp, $t$")

plt.ylabel("probability estimate" )
plt.title( "Posterior probability estimates given temp. $t$" );


The 95% credible interval, or 95% CI, painted in purple, represents the interval, for each temperature, that contains 95% of the distribution. For example, at 65 degrees, we can be 95% sure that the probability of defect lies between 0.25 and 0.75.

More generally, we can see that as the temperature nears 60 degrees, the CI's spread out over [0,1] quickly. As we pass 70 degrees, the CI's tighten again. This can give us insight about how to proceed next: we should probably test more O-rings around 60-65 temperature to get a better estimate of probabilities in that range. Similarly, when reporting to scientists your estimates, you should be very cautious about simply telling them the expected probability, as we can see this does not reflect how wide the posterior distribution is.

### What about the day of the Challenger disaster?¶

On the day of the Challenger disaster, the outside temperature was 31 degrees Fahrenheit. What is the posterior distribution of a defect occurring, given this temperature? The distribution is plotted below. It looks almost guaranteed that the Challenger was going to be subject to defective O-rings.

In [71]:
figsize(12.5, 2.5)

prob_31 = logistic( 31, beta_samples, alpha_samples )

plt.xlim( 0.995, 1)
plt.hist( prob_31, bins = 1000, normed = True, histtype='stepfilled' )
plt.title( "Posterior distribution of probability of defect, given $t = 31$")
plt.xlabel( "probability of defect occuring in O-ring" );


### Is our model appropriate?¶

The skeptical reader will say "You deliberately chose the logistic function for $p(t)$ and the specific priors. Perhaps other functions or priors will give different results. How do I know I have chosen a good model?" This is absolutely true. To consider an extreme situation, what if I had chosen the function $p(t) = 1,\; \forall t$, which guarantees a defect always occurring: I would have again predicted disaster on January 28th. Yet this is clearly a poorly chosen model. On the other hand, if I did choose the logistic function for $p(t)$, but specified all my priors to be very tight around 0, likely we would have very different posterior distributions. How do we know our model is an expression of the data? This encourages us to measure the model's goodness of fit.

We can think: how can we test whether our model is a bad fit? An idea is to compare observed data (which if we recall is a fixed stochastic variable) with artificial dataset which we can simulate. The rational is that if the simulated dataset does not appear similar, statistically, to the observed dataset, then likely our model is not accurately represented the observed data.

Previously in this Chapter, we simulated artificial dataset for the SMS example. To do this, we sampled values from the priors. We saw how varied the resulting datasets looked like, and rarely did they mimic our observed dataset. In the current example, we should sample from the posterior distributions to create very plausible datasets. Luckily, our Bayesian framework makes this very easy. We only need to create a new Stochastic variable, that is exactly the same as our variable that stored the observations, but minus the observations themselves. If you recall, our Stochastic variable that stored our observed data was:

observed = mc.Bernoulli( "bernoulli_obs", p, value = D, observed=True)



Hence we create:

simulated_data = mc.Bernoulli("simulation_data", p )



Let's simulate 10 000:

In [57]:
simulated = mc.Bernoulli( "bernoulli_sim", p)
N = 10000

mcmc = mc.MCMC( [simulated, alpha, beta, observed ] )
mcmc.sample( N )

[****************100%******************]  10000 of 10000 complete

In [58]:
figsize(12.5, 5)

simulations = mcmc.trace("bernoulli_sim")[:]
print simulations.shape

plt.title( "Simulated dataset using posterior parameters")
figsize( 12.5, 6)
for i in range(4):
ax = subplot( 4, 1, i+1)
plt.scatter( temperature, simulations[1000*i,:], color = "k",
s = 50, alpha = 0.6 );

(10000L, 23L)


Note that the above plots are different (if you can think of a cleaner way to present this, please send a pull request and answer here!).

We wish to assess how good our model is. "Good" is a subjective term of course, so results must be relative to other models.

We will be doing this graphically as well, which may seem like an even less objective method. The alternative is to use Bayesian p-values. These are still subjective, as the proper cutoff between good and bad is arbitrary. Gelman emphasises that the graphical tests are more illuminating [7] than p-value tests. We agree.

The following graphical test is a novel data-viz approach to logistic regression. The plots are called separation plots[8]. For a suite of models we wish to compare, each model is plotted on an individual separation plot. I leave most of the technical details about separation plots to the very accessible original paper, but I'll summarize their use here.

For each model, we calculate the proportion of times the posterior simulation proposed a value of 1 for a particular temperature, i.e. compute $P( \;\text{Defect} = 1 | t, \alpha, \beta )$ by averaging. This gives us the posterior probability of a defect at each data point in our dataset. For example, for the model we used above:

In [65]:
posterior_probability = simulations.mean(axis=0)
print "posterior prob of defect | realized defect "
for i in range( len(D) ):
print "%.2f                     |   %d"%(posterior_probability[i], D[i])

posterior prob of defect | realized defect
0.44                     |   0
0.26                     |   1
0.29                     |   0
0.35                     |   0
0.39                     |   0
0.19                     |   0
0.16                     |   0
0.26                     |   0
0.82                     |   1
0.59                     |   1
0.26                     |   1
0.07                     |   0
0.39                     |   0
0.91                     |   1
0.39                     |   0
0.12                     |   0
0.26                     |   0
0.04                     |   0
0.10                     |   0
0.06                     |   0
0.12                     |   1
0.10                     |   0
0.79                     |   1


Next we sort each column by the posterior probabilities:

In [66]:
ix = np.argsort( posterior_probability )
print "probb | defect "
for i in range( len(D) ):
print "%.2f  |   %d"%(posterior_probability[ix[i]], D[ix[i]])

probb | defect
0.04  |   0
0.06  |   0
0.07  |   0
0.10  |   0
0.10  |   0
0.12  |   1
0.12  |   0
0.16  |   0
0.19  |   0
0.26  |   1
0.26  |   0
0.26  |   0
0.26  |   1
0.29  |   0
0.35  |   0
0.39  |   0
0.39  |   0
0.39  |   0
0.44  |   0
0.59  |   1
0.79  |   1
0.82  |   1
0.91  |   1


We can present the above data better in a figure: I've wrapped this up into a separation_plot function.

In [80]:
from separation_plot import separation_plot

figsize( 11., 1.5 )

separation_plot(posterior_probability, D )


The snaking-line is the sorted probabilities, blue bars denote defects, and empty space (or grey bars for the optimistic readers) denote non-defects. As the probability rises, we see more and more defects occur. On the right hand side, the plot suggests that as the posterior probability is large (line close to 1), then more defects are realized. This is good behaviour. Ideally, all the blue bars should be close to the right-hand side, and deviations from this reflect missed predictions.

The black vertical line is the expected number of defects we should observe, given this model. This allows the user to see how the total number of events predicted by the model compares to the actual number of events in the data.

It is much more informative to compare this to separation plots for other models. Below we compare our model (top) versus three others:

1. the perfect model, which predicts the posterior probability to be equal 1 if a defect did occur.
2. a completely random model, which predicts random probabilities regardless of temperature.
3. a constant model: where $P(D = 1 \; | \; t) = c, \;\; \forall t$. The best choice for $c$ is the observed frequency of defects, in this case 7/23.
In [91]:
figsize( 11., 1.25 )

# our temperature-dependent model
separation_plot(posterior_probability, D )
plt.title("Temperature-dependent model")

# perfect model
# i.e. the probability of defect is equal to if a defect occurred or not.
p = D
separation_plot( p, D)
plt.title("Perfect model")

# random predictions
p = np.random.rand( 23 )
separation_plot( p, D)
plt.title("Random model")

# constant model
constant_prob = 7./23*np.ones(23)
separation_plot(constant_prob, D )
plt.title("Constant-prediction model");


In the random model, we can see that as the probability increases there is no clustering of defects to the right-hand side. Similarly for the constant model.

The perfect model, the probability line is not well shown, as it is stuck to the bottom and top of the figure. Of course the perfect model is only for demonstration, and we cannot infer any scientific inference from it.

##### Exercises¶

1. Try putting in extreme values for our observations in the cheating example. What happens if we observe 25 affirmative responses? 10? 50?

2. Try plotting $\alpha$ samples versus $\beta$ samples. Why might the resulting plot look like this?

In [90]:
#type your code here.
figsize(12.5, 4 )

plt.scatter( alpha_samples, beta_samples, alpha = 0.1 )
plt.title( "Why does the plot look like this?" )
plt.xlabel( r"$\alpha$")
plt.ylabel( r"$\beta$");


### References¶

• [1] Dalal, Fowlkes and Hoadley (1989),JASA, 84, 945-957.
• [2] German Rodriguez. Datasets. In WWS509. Retrieved 30/01/2013, from http://data.princeton.edu/wws509/datasets/#smoking.
• [3] McLeish, Don, and Cyntha Struthers. STATISTICS 450/850 Estimation and Hypothesis Testing. Winter 2012. Waterloo, Ontario: 2012. Print.
• [4] Fonnesbeck, Christopher. "Building Models." PyMC-Devs. N.p., n.d. Web. 26 Feb 2013. http://pymc-devs.github.com/pymc/modelbuilding.html.
• [5] Cronin, Beau. "Why Probabilistic Programming Matters." 24 Mar 2013. Google, Online Posting to Google . Web. 24 Mar. 2013. https://plus.google.com/u/0/107971134877020469960/posts/KpeRdJKR6Z1.
• [6] S.P. Brooks, E.A. Catchpole, and B.J.T. Morgan. Bayesian animal survival estimation. Statistical Science, 15: 357–376, 2000
• [7] Gelman, Andrew. "Philosophy and the practice of Bayesian statistics." British Journal of Mathematical and Statistical Psychology. (2012): n. page. Web. 2 Apr. 2013.
• [8] Greenhill, Brian, Michael D. Ward, and Audrey Sacks. "The Separation Plot: A New Visual Method for Evaluating the Fit of Binary Models." American Journal of Political Science. 55.No.4 (2011): n. page. Web. 2 Apr. 2013.
In [1]:
from IPython.core.display import HTML
def css_styling():