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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Getting started with PyMC3
Authors
Step1:  Here is what the simulated data look like. We use the pylab module from the plotting library matplotlib.
Step2:  Model Specification
Specifying this model in PyMC3 is straightforward because the syntax is as close to the statistical notation. For the most part, each line of Python code corresponds to a line in the model notation above. 
First, we import the components we will need from PyMC.
Step3:  Now we build our model, which we will present in full first, then explain each part line-by-line.
Step4:  The first line,
python
basic_model = Model()
creates a new Model object which is a container for the model random variables.
Following instantiation of the model, the subsequent specification of the model components is performed inside a  with statement
Step5:  Having defined the priors, the next statement creates the expected value mu of the outcomes, specifying the linear relationship
Step6:  By default, find_MAP uses the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization algorithm to find the maximum of the log-posterior but also allows selection of other optimization algorithms from the scipy.optimize module. For example, below we use Powell's method to find the MAP.
Step7:  It is important to note that the MAP estimate is not always reasonable, especially if the mode is at an extreme. This can be a subtle issue; with high dimensional posteriors, one can have areas of extremely high density but low total probability because the volume is very small. This will often occur in hierarchical models with the variance parameter for the random effect. If the individual group means are all the same, the posterior will have near infinite density if the scale parameter for the group means is almost zero, even though the probability of such a small scale parameter will be small since the group means must be extremely close together. 
Most techniques for finding the MAP estimate also only find a local optimum (which is often good enough), but can fail badly for multimodal posteriors if the different modes are meaningfully different.
Sampling methods
Though finding the MAP is a fast and easy way of obtaining estimates of the unknown model parameters, it is limited because there is no associated estimate of uncertainty produced with the MAP estimates. Instead, a simulation-based approach such as Markov chain Monte Carlo (MCMC) can be used to obtain a Markov chain of values that, given the satisfaction of certain conditions, are indistinguishable from samples from the posterior distribution. 
To conduct MCMC sampling to generate posterior samples in PyMC3, we specify a step method object that corresponds to a particular MCMC algorithm, such as Metropolis, Slice sampling, or the No-U-Turn Sampler (NUTS). PyMC3's step_methods submodule contains the following samplers
Step8:  The sample function runs the step method(s) assigned (or passed) to it for the given number of iterations and returns a Trace object containing the samples collected, in the order they were collected. The trace object can be queried in a similar way to a dict containing a map from variable names to numpy.arrays. The first dimension of the array is the sampling index and the later dimensions match the shape of the variable. We can see the last 5 values for the alpha variable as follows
Step9:  If we wanted to use the slice sampling algorithm to sigma instead of NUTS (which was assigned automatically), we could have specified this as the step argument for sample.
Step10:  Posterior analysis
PyMC3 provides plotting and summarization functions for inspecting the sampling output. A simple posterior plot can be created using traceplot.
Step11:  The left column consists of a smoothed histogram (using kernel density estimation) of the marginal posteriors of each stochastic random variable while the right column contains the samples of the Markov chain plotted in sequential order. The beta variable, being vector-valued, produces two histograms and two sample traces, corresponding to both predictor coefficients.
In addition, the summary function provides a text-based output of common posterior statistics
Step12:  Case study 1
Step13:  Model Specification
As with the linear regression example, specifying the model in PyMC3 mirrors its statistical specification. This model employs several new distributions
Step14:  Notice that we transform the log volatility process s into the volatility process by exp(-2*s). Here, exp is a Theano function, rather than the corresponding function in NumPy; Theano provides a large subset of the mathematical functions that NumPy does.
Also note that we have declared the Model name sp500_model in the first occurrence of the context manager, rather than splitting it into two lines, as we did for the first example.
Fitting
Before we draw samples from the posterior, it is prudent to find a decent starting value by finding a point of relatively high probability. For this model, the full maximum a posteriori (MAP) point over all variables is degenerate and has infinite density. But, if we fix log_sigma and nu it is no longer degenerate, so we find the MAP with respect only to the volatility process s keeping log_sigma and nu constant at their default values (remember that we set testval=.1 for sigma). We use the Limited-memory BFGS (L-BFGS) optimizer, which is provided by the scipy.optimize package, as it is more efficient for high dimensional functions and we have 400 stochastic random variables (mostly from s).
To achieve good convergence with NUTS, it is critical to find a good scaling. The MAP as found by find_MAP() is in our experience a poor choice. Instead, we run automatic differentiation variational inference (ADVI) to get a variational estimate of the posterior mean as well as the posterior standard deviations. We can use these estimates to initialize NUTS to achieve faster sampling due to better scaling.
Step15:  We can check our samples by looking at the traceplot for nu and sigma.
Step16:  Finally we plot the distribution of volatility paths by plotting many of our sampled volatility paths on the same graph. Each is rendered partially transparent (via the alpha argument in Matplotlib's plot function) so the regions where many paths overlap are shaded more darkly.
Step17:  As you can see, the model correctly infers the increase in volatility during the 2008 financial crash. Moreover, note that this model is quite complex because of its high dimensionality and dependency-structure in the random walk distribution. NUTS as implemented in PyMC3, however, correctly infers the posterior distribution with ease.
Case study 2
Step18:  Occurrences of disasters in the time series is thought to follow a Poisson process with a large rate parameter in the early part of the time series, and from one with a smaller rate in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations.
In our model, 
$$ 
\begin{aligned}
  D_t &\sim \text{Pois}(r_t), r_t= \begin{cases} 
   l, & \text{if } t \lt s \
   e, & \text{if } t \ge s 
   \end{cases} \
  s &\sim \text{Unif}(t_l, t_h)\       
  e &\sim \text{exp}(1)\
  l &\sim \text{exp}(1)  
\end{aligned}
$$
 the parameters are defined as follows
Step19:  The logic for the rate random variable,
python
rate = switch(switchpoint &gt;= year, early_rate, late_rate)
is implemented using switch, a Theano function that works like an if statement. It uses the first argument to switch between the next two arguments.
Missing values are handled transparently by passing a MaskedArray or a pandas.DataFrame with NaN values to the observed argument when creating an observed stochastic random variable. Behind the scenes, another random variable, disasters.missing_values is created to model the missing values. All we need to do to handle the missing values is ensure we sample this random variable as well.
Unfortunately because they are discrete variables and thus have no meaningful gradient, we cannot use NUTS for sampling switchpoint or the missing disaster observations. Instead, we will sample using a Metroplis step method, which implements adaptive Metropolis-Hastings, because it is designed to handle discrete values.
We sample with both samplers at once by passing them to the sample function in a list. Each new sample is generated by first applying step1 then step2.
Step20:  In the trace plot below we can see that there's about a 10 year span that's plausible for a significant change in safety, but a 5 year span that contains most of the probability mass. The distribution is jagged because of the jumpy relationship between the year switchpoint and the likelihood  and not due to sampling error.
Step21:  Arbitrary deterministics
Due to its reliance on Theano, PyMC3 provides many mathematical functions and operators for transforming random variables into new random variables. However, the library of functions in Theano is not exhaustive, therefore Theano and PyMC3 provide functionality for creating arbitrary Theano functions in pure Python, and including these functions in PyMC models. This is supported with the as_op function decorator.
Theano needs to know the types of the inputs and outputs of a function, which are specified for as_op by itypes for inputs and otypes for outputs. The Theano documentation includes an overview of the available types.
Step22:  An important drawback of this approach is that it is not possible for theano to inspect these functions in order to compute the gradient required for the Hamiltonian-based samplers. Therefore, it is not possible to use the HMC or NUTS samplers for a model that uses such an operator. However, it is possible to add a gradient if we inherit from theano.Op instead of using as_op. The PyMC example set includes a more elaborate example of the usage of as_op.
Arbitrary distributions
Similarly, the library of statistical distributions in PyMC3 is not exhaustive, but PyMC allows for the creation of user-defined functions for an arbitrary probability distribution. For simple statistical distributions, the DensityDist function takes as an argument any function that calculates a log-probability $log(p(x))$. This function may employ other random variables in its calculation. Here is an example inspired by a blog post by Jake Vanderplas on which priors to use for a linear regression (Vanderplas, 2014). 
```python
import theano.tensor as T
from pymc3 import DensityDist, Uniform
with Model() as model
Step23:  Generalized Linear Models
Generalized Linear Models (GLMs) are a class of flexible models that are widely used to estimate regression relationships between a single outcome variable and one or multiple predictors. Because these models are so common, PyMC3 offers a glm submodule that allows flexible creation of various GLMs with an intuitive R-like syntax that is implemented via the patsy module.
The glm submodule requires data to be included as a pandas DataFrame. Hence, for our linear regression example
Step24:  The model can then be very concisely specified in one line of code.
Step25:  The error distribution, if not specified via the family argument, is assumed to be normal. In the case of logistic regression, this can be modified by passing in a Binomial family object.
Step26:  Backends
PyMC3 has support for different ways to store samples during and after sampling, called backends, including in-memory (default), text file, and SQLite. These can be found in pymc.backends
Step27:  The stored trace can then later be loaded using the load command
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Python Code:
import numpy as np
import matplotlib.pyplot as plt
# Initialize random number generator
np.random.seed(123)
# True parameter values
alpha, sigma = 1, 1
beta = [1, 2.5]
# Size of dataset
size = 100
# Predictor variable
X1 = np.random.randn(size)
X2 = np.random.randn(size) * 0.2
# Simulate outcome variable
Y = alpha + beta[0]*X1 + beta[1]*X2 + np.random.randn(size)*sigma
Explanation: Getting started with PyMC3
Authors: John Salvatier, Thomas V. Wiecki, Christopher Fonnesbeck
Note: This text is taken from the PeerJ CS publication on PyMC3.
Abstract
Probabilistic Programming allows for automatic Bayesian inference on user-defined probabilistic models. Recent advances in Markov chain Monte Carlo (MCMC) sampling allow inference on increasingly complex models. This class of MCMC, known as Hamliltonian Monte Carlo, requires gradient information which is often not readily available. PyMC3 is a new open source Probabilistic Programming framework written in Python that uses Theano to compute gradients via automatic differentiation as well as compile probabilistic programs on-the-fly to C for increased speed. Contrary to other Probabilistic Programming languages, PyMC3 allows model specification directly in Python code. The lack of a domain specific language allows for great flexibility and direct interaction with the model. This paper is a tutorial-style introduction to this software package.
Introduction
Probabilistic programming (PP) allows flexible specification of Bayesian statistical models in code. PyMC3 is a new, open-source PP framework with an intuitive and readable, yet powerful, syntax that is close to the natural syntax statisticians use to describe models. It features next-generation Markov chain Monte Carlo (MCMC) sampling algorithms such as the No-U-Turn Sampler (NUTS; Hoffman, 2014), a self-tuning variant of Hamiltonian Monte Carlo (HMC; Duane, 1987). This class of samplers works well on high dimensional and complex posterior distributions and allows many complex models to be fit without specialized knowledge about fitting algorithms. HMC and NUTS take advantage of gradient information from the likelihood to achieve much faster convergence than traditional sampling methods, especially for larger models. NUTS also has several self-tuning strategies for adaptively setting the tunable parameters of Hamiltonian Monte Carlo, which means you usually don't need to have specialized knowledge about how the algorithms work. PyMC3, Stan (Stan Development Team, 2014), and the LaplacesDemon package for R are currently the only PP packages to offer HMC.
Probabilistic programming in Python confers a number of advantages including multi-platform compatibility, an expressive yet clean and readable syntax, easy integration with other scientific libraries, and extensibility via C, C++, Fortran or Cython. These features make it relatively straightforward to write and use custom statistical distributions, samplers and transformation functions, as required by Bayesian analysis.
While most of PyMC3's user-facing features are written in pure Python, it leverages Theano (Bergstra et al., 2010) to transparently transcode models to C and compile them to machine code, thereby boosting performance. Theano is a library that allows expressions to be defined using generalized vector data structures called tensors, which are tightly integrated with the popular NumPy ndarray data structure, and similarly allow for broadcasting and advanced indexing, just as NumPy arrays do. Theano also automatically optimizes the likelihood's computational graph for speed and provides simple GPU integration.
Here, we present a primer on the use of PyMC3 for solving general Bayesian statistical inference and prediction problems. We will first see the basics of how to use PyMC3, motivated by a simple example: installation, data creation, model definition, model fitting and posterior analysis. Then we will cover two case studies and use them to show how to define and fit more sophisticated models. Finally we will show how to extend PyMC3 and discuss other useful features: the Generalized Linear Models subpackage, custom distributions, custom transformations and alternative storage backends.
Installation
Running PyMC3 requires a working Python interpreter, either version 2.7 (or more recent) or 3.4 (or more recent); we recommend that new users install version 3.4. A complete Python installation for Mac OSX, Linux and Windows can most easily be obtained by downloading and installing the free Anaconda Python Distribution by ContinuumIO. 
PyMC3 can be installed using pip (https://pip.pypa.io/en/latest/installing.html):
pip install git+https://github.com/pymc-devs/pymc3
PyMC3 depends on several third-party Python packages which will be automatically installed when installing via pip. The four required dependencies are: Theano, NumPy, SciPy, and Matplotlib. 
To take full advantage of PyMC3, the optional dependencies Pandas and Patsy should also be installed. These are not automatically installed, but can be installed by:
pip install patsy pandas
The source code for PyMC3 is hosted on GitHub at https://github.com/pymc-devs/pymc3 and is distributed under the liberal Apache License 2.0. On the GitHub site, users may also report bugs and other issues, as well as contribute code to the project, which we actively encourage.
A Motivating Example: Linear Regression
To introduce model definition, fitting and posterior analysis, we first consider a simple Bayesian linear regression model with normal priors for the parameters. We are interested in predicting outcomes $Y$ as normally-distributed observations with an expected value $\mu$ that is a linear function of two predictor variables, $X_1$ and $X_2$.
$$\begin{aligned} 
Y  &\sim \mathcal{N}(\mu, \sigma^2) \
\mu &= \alpha + \beta_1 X_1 + \beta_2 X_2
\end{aligned}$$
where $\alpha$ is the intercept, and $\beta_i$ is the coefficient for covariate $X_i$, while $\sigma$ represents the observation error. Since we are constructing a Bayesian model, the unknown variables in the model must be assigned a prior distribution. We choose zero-mean normal priors with variance of 100 for both regression coefficients, which corresponds to weak information regarding the true parameter values. We choose a half-normal distribution (normal distribution bounded at zero) as the prior for $\sigma$.
$$\begin{aligned} 
\alpha &\sim \mathcal{N}(0, 100) \
\beta_i &\sim \mathcal{N}(0, 100) \
\sigma &\sim \lvert\mathcal{N}(0, 1){\rvert}
\end{aligned}$$
Generating data
We can simulate some artificial data from this model using only NumPy's random module, and then use PyMC3 to try to recover the corresponding parameters. We are intentionally generating the data to closely correspond the PyMC3 model structure.
End of explanation
%matplotlib inline 
fig, axes = plt.subplots(1, 2, sharex=True, figsize=(10,4))
axes[0].scatter(X1, Y)
axes[1].scatter(X2, Y)
axes[0].set_ylabel('Y'); axes[0].set_xlabel('X1'); axes[1].set_xlabel('X2');
Explanation: Here is what the simulated data look like. We use the pylab module from the plotting library matplotlib.
End of explanation
from pymc3 import Model, Normal, HalfNormal
Explanation: Model Specification
Specifying this model in PyMC3 is straightforward because the syntax is as close to the statistical notation. For the most part, each line of Python code corresponds to a line in the model notation above. 
First, we import the components we will need from PyMC.
End of explanation
basic_model = Model()
with basic_model:
    
    # Priors for unknown model parameters
    alpha = Normal('alpha', mu=0, sd=10)
    beta = Normal('beta', mu=0, sd=10, shape=2)
    sigma = HalfNormal('sigma', sd=1)
    
    # Expected value of outcome
    mu = alpha + beta[0]*X1 + beta[1]*X2
    
    # Likelihood (sampling distribution) of observations
    Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=Y)
Explanation: Now we build our model, which we will present in full first, then explain each part line-by-line.
End of explanation
help(Normal) #try help(Model), help(Uniform) or help(basic_model)
Explanation: The first line,
python
basic_model = Model()
creates a new Model object which is a container for the model random variables.
Following instantiation of the model, the subsequent specification of the model components is performed inside a  with statement:
python
with basic_model:
This creates a context manager, with our basic_model as the context, that includes all statements until the indented block ends. This means all PyMC3 objects introduced in the indented code block below the with statement are added to the model behind the scenes. Absent this context manager idiom, we would be forced to manually associate each of the variables with basic_model right after we create them. If you try to create a new random variable without a with model: statement, it will raise an error since there is no obvious model for the variable to be added to.
The first three statements in the context manager:
python
alpha = Normal('alpha', mu=0, sd=10)
beta = Normal('beta', mu=0, sd=10, shape=2)
sigma = HalfNormal('sigma', sd=1)
create a stochastic random variables with a Normal prior distributions for the regression coefficients with a mean of 0 and standard deviation of 10 for the regression coefficients, and a half-normal distribution for the standard deviation of the observations, $\sigma$. These are stochastic because their values are partly determined by its parents in the dependency graph of random variables, which for priors are simple constants, and partly random (or stochastic). 
We call the Normal constructor to create a random variable to use as a normal prior. The first argument is always the name of the random variable, which should almost always match the name of the Python variable being assigned to, since it sometimes used to retrieve the variable from the model for summarizing output. The remaining required arguments for a stochastic object are the parameters, in this case mu, the mean, and sd, the standard deviation, which we assign hyperparameter values for the model. In general, a distribution's parameters are values that determine the location, shape or scale of the random variable, depending on the parameterization of the distribution. Most commonly used distributions, such as Beta, Exponential, Categorical, Gamma, Binomial and many others, are available in PyMC3.
The beta variable has an additional shape argument to denote it as a vector-valued parameter of size 2. The shape argument is available for all distributions and specifies the length or shape of the random variable, but is optional for scalar variables, since it defaults to a value of one. It can be an integer, to specify an array, or a tuple, to specify a multidimensional array (e.g. shape=(5,7) makes random variable that takes on 5 by 7 matrix values). 
Detailed notes about distributions, sampling methods and other PyMC3 functions are available via the help function.
End of explanation
from pymc3 import find_MAP
map_estimate = find_MAP(model=basic_model)
    
print(map_estimate)
Explanation: Having defined the priors, the next statement creates the expected value mu of the outcomes, specifying the linear relationship:
python
mu = alpha + beta[0]*X1 + beta[1]*X2
This creates a deterministic random variable, which implies that its value is completely determined by its parents' values. That is, there is no uncertainty beyond that which is inherent in the parents' values. Here, mu is just the sum of the intercept alpha and the two products of the coefficients in beta and the predictor variables, whatever their values may be. 
PyMC3 random variables and data can be arbitrarily added, subtracted, divided, multiplied together and indexed-into to create new random variables. This allows for great model expressivity. Many common mathematical functions like sum, sin, exp and linear algebra functions like dot (for inner product) and inv (for inverse) are also provided. 
The final line of the model, defines Y_obs, the sampling distribution of the outcomes in the dataset.
python
Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=Y)
This is a special case of a stochastic variable that we call an observed stochastic, and represents the data likelihood of the model. It is identical to a standard stochastic, except that its observed argument, which passes the data to the variable, indicates that the values for this variable were observed, and should not be changed by any fitting algorithm applied to the model. The data can be passed in the form of either a numpy.ndarray or pandas.DataFrame object.
Notice that, unlike for the priors of the model, the parameters for the normal distribution of Y_obs are not fixed values, but rather are the deterministic object mu and the stochastic sigma. This creates parent-child relationships between the likelihood and these two variables.
Model fitting
Having completely specified our model, the next step is to obtain posterior estimates for the unknown variables in the model. Ideally, we could calculate the posterior estimates analytically, but for most non-trivial models, this is not feasible. We will consider two approaches, whose appropriateness depends on the structure of the model and the goals of the analysis: finding the maximum a posteriori (MAP) point using optimization methods, and computing summaries based on samples drawn from the posterior distribution using Markov Chain Monte Carlo (MCMC) sampling methods.
Maximum a posteriori methods
The maximum a posteriori (MAP) estimate for a model, is the mode of the posterior distribution and is generally found using numerical optimization methods. This is often fast and easy to do, but only gives a point estimate for the parameters and can be biased if the mode isn't representative of the distribution. PyMC3 provides this functionality with the find_MAP function.
Below we find the MAP for our original model. The MAP is returned as a parameter point, which is always represented by a Python dictionary of variable names to NumPy arrays of parameter values.
End of explanation
from scipy import optimize
map_estimate = find_MAP(model=basic_model, fmin=optimize.fmin_powell)
    
print(map_estimate)
Explanation: By default, find_MAP uses the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization algorithm to find the maximum of the log-posterior but also allows selection of other optimization algorithms from the scipy.optimize module. For example, below we use Powell's method to find the MAP.
End of explanation
from pymc3 import NUTS, sample
from scipy import optimize
with basic_model:
    
    # obtain starting values via MAP
    start = find_MAP(fmin=optimize.fmin_powell)
    
    # draw 2000 posterior samples
    trace = sample(2000, start=start) 
Explanation: It is important to note that the MAP estimate is not always reasonable, especially if the mode is at an extreme. This can be a subtle issue; with high dimensional posteriors, one can have areas of extremely high density but low total probability because the volume is very small. This will often occur in hierarchical models with the variance parameter for the random effect. If the individual group means are all the same, the posterior will have near infinite density if the scale parameter for the group means is almost zero, even though the probability of such a small scale parameter will be small since the group means must be extremely close together. 
Most techniques for finding the MAP estimate also only find a local optimum (which is often good enough), but can fail badly for multimodal posteriors if the different modes are meaningfully different.
Sampling methods
Though finding the MAP is a fast and easy way of obtaining estimates of the unknown model parameters, it is limited because there is no associated estimate of uncertainty produced with the MAP estimates. Instead, a simulation-based approach such as Markov chain Monte Carlo (MCMC) can be used to obtain a Markov chain of values that, given the satisfaction of certain conditions, are indistinguishable from samples from the posterior distribution. 
To conduct MCMC sampling to generate posterior samples in PyMC3, we specify a step method object that corresponds to a particular MCMC algorithm, such as Metropolis, Slice sampling, or the No-U-Turn Sampler (NUTS). PyMC3's step_methods submodule contains the following samplers: NUTS, Metropolis, Slice, HamiltonianMC, and BinaryMetropolis. These step methods can be assigned manually, or assigned automatically by PyMC3. Auto-assignment is based on the attributes of each variable in the model. In general:
Binary variables will be assigned to BinaryMetropolis
Discrete variables will be assigned to Metropolis
Continuous variables will be assigned to NUTS
Auto-assignment can be overriden for any subset of variables by specifying them manually prior to sampling.
Gradient-based sampling methods
PyMC3 has the standard sampling algorithms like adaptive Metropolis-Hastings and adaptive slice sampling, but PyMC3's most capable step method is the No-U-Turn Sampler. NUTS is especially useful on models that have many continuous parameters, a situation where other MCMC algorithms work very slowly. It takes advantage of information about where regions of higher probability are, based on the gradient of the log posterior-density. This helps it achieve dramatically faster convergence on large problems than traditional sampling methods achieve. PyMC3 relies on Theano to analytically compute model gradients via automatic differentiation of the posterior density. NUTS also has several self-tuning strategies for adaptively setting the tunable parameters of Hamiltonian Monte Carlo. For random variables that are undifferentiable (namely, discrete variables) NUTS cannot be used, but it may still be used on the differentiable variables in a model that contains undifferentiable variables. 
NUTS requires a scaling matrix parameter, which is analogous to the variance parameter for the jump proposal distribution in Metropolis-Hastings, although NUTS uses it somewhat differently. The matrix gives the rough shape of the distribution so that NUTS does not make jumps that are too large in some directions and too small in other directions. It is important to set this scaling parameter to a reasonable value to facilitate efficient sampling. This is especially true for models that have many unobserved stochastic random variables or models with highly non-normal posterior distributions. Poor scaling parameters will slow down NUTS significantly, sometimes almost stopping it completely. A reasonable starting point for sampling can also be important for efficient sampling, but not as often.
Fortunately NUTS can often make good guesses for the scaling parameters. If you pass a point in parameter space (as a dictionary of variable names to parameter values, the same format as returned by find_MAP) to NUTS, it will look at the local curvature of the log posterior-density (the diagonal of the Hessian matrix) at that point to make a guess for a good scaling vector, which often results in a good value. The MAP estimate is often a good point to use to initiate sampling. It is also possible to supply your own vector or scaling matrix to NUTS, though this is a more advanced use. If you wish to modify a Hessian at a specific point to use as your scaling matrix or vector, you can use find_hessian or find_hessian_diag.
For our basic linear regression example in basic_model, we will use NUTS to sample 2000 draws from the posterior using the MAP as the starting point and scaling point. This must also be performed inside the context of the model.
End of explanation
trace['alpha'][-5:]
Explanation: The sample function runs the step method(s) assigned (or passed) to it for the given number of iterations and returns a Trace object containing the samples collected, in the order they were collected. The trace object can be queried in a similar way to a dict containing a map from variable names to numpy.arrays. The first dimension of the array is the sampling index and the later dimensions match the shape of the variable. We can see the last 5 values for the alpha variable as follows:
End of explanation
from pymc3 import Slice
with basic_model:
    
    # obtain starting values via MAP
    start = find_MAP(fmin=optimize.fmin_powell)
    
    # instantiate sampler
    step = Slice(vars=[sigma]) 
    
    # draw 5000 posterior samples
    trace = sample(5000, step=step, start=start)   
Explanation: If we wanted to use the slice sampling algorithm to sigma instead of NUTS (which was assigned automatically), we could have specified this as the step argument for sample.
End of explanation
from pymc3 import traceplot
traceplot(trace);
Explanation: Posterior analysis
PyMC3 provides plotting and summarization functions for inspecting the sampling output. A simple posterior plot can be created using traceplot.
End of explanation
from pymc3 import summary
summary(trace)
Explanation: The left column consists of a smoothed histogram (using kernel density estimation) of the marginal posteriors of each stochastic random variable while the right column contains the samples of the Markov chain plotted in sequential order. The beta variable, being vector-valued, produces two histograms and two sample traces, corresponding to both predictor coefficients.
In addition, the summary function provides a text-based output of common posterior statistics:
End of explanation
try:
    from pandas_datareader import data
except ImportError:
    !pip install pandas-datareader
    from pandas_datareader import data
import pandas as pd
returns = data.get_data_yahoo('SPY', start='2008-5-1', end='2009-12-1')['Adj Close'].pct_change()
print(len(returns))
returns.plot(figsize=(10, 6))
plt.ylabel('daily returns in %');
Explanation: Case study 1: Stochastic volatility
We present a case study of stochastic volatility, time varying stock market volatility, to illustrate PyMC3's use in addressing a more realistic problem. The distribution of market returns is highly non-normal, which makes sampling the volatilities significantly more difficult. This example has 400+ parameters so using common sampling algorithms like Metropolis-Hastings would get bogged down, generating highly autocorrelated samples. Instead, we use NUTS, which is dramatically more efficient.
The Model
Asset prices have time-varying volatility (variance of day over day returns). In some periods, returns are highly variable, while in others they are very stable. Stochastic volatility models address this with a latent volatility variable, which changes over time. The following model is similar to the one described in the NUTS paper (Hoffman 2014, p. 21).
$$\begin{aligned} 
  \sigma &\sim exp(50) \
  \nu &\sim exp(.1) \
  s_i &\sim \mathcal{N}(s_{i-1}, \sigma^{-2}) \
  log(y_i) &\sim t(\nu, 0, exp(-2 s_i))
\end{aligned}$$
Here, $y$ is the daily return series which is modeled with a Student-t distribution with an unknown degrees of freedom parameter, and a scale parameter determined by a latent process $s$. The individual $s_i$ are the individual daily log volatilities in the latent log volatility process. 
The Data
Our data consist of daily returns of the S&P 500 during the 2008 financial crisis. Here, we use pandas-datareader to obtain the price data from Yahoo!-Finance; it can be installed with pip install pandas-datareader.
End of explanation
from pymc3 import Exponential, StudentT, Deterministic
from pymc3.math import exp
from pymc3.distributions.timeseries import GaussianRandomWalk
with Model() as sp500_model:
    
    nu = Exponential('nu', 1./10, testval=5.)
    
    sigma = Exponential('sigma', 1./.02, testval=.1)
    
    s = GaussianRandomWalk('s', sigma**-2, shape=len(returns))
    
    volatility_process = Deterministic('volatility_process', exp(-2*s))
    
    r = StudentT('r', nu, lam=1/volatility_process, observed=returns)
Explanation: Model Specification
As with the linear regression example, specifying the model in PyMC3 mirrors its statistical specification. This model employs several new distributions: the Exponential distribution for the $ \nu $ and $\sigma$ priors, the Student-T (StudentT) distribution for distribution of returns, and the GaussianRandomWalk for the prior for the latent volatilities.   
In PyMC3, variables with purely positive priors like Exponential are transformed with a log transform. This makes sampling more robust. Behind the scenes, a variable in the unconstrained space (named "variableName_log") is added to the model for sampling. In this model this happens behind the scenes for both the degrees of freedom, nu, and the scale parameter for the volatility process, sigma, since they both have exponential priors. Variables with priors that constrain them on two sides, like Beta or Uniform, are also transformed to be unconstrained but with a log odds transform. 
Although, unlike model specification in PyMC2, we do not typically provide starting points for variables at the model specification stage, we can also provide an initial value for any distribution (called a "test value") using the testval argument. This overrides the default test value for the distribution (usually the mean, median or mode of the distribution), and is most often useful if some values are illegal and we want to ensure we select a legal one. The test values for the distributions are also used as a starting point for sampling and optimization by default, though this is easily overriden. 
The vector of latent volatilities s is given a prior distribution by GaussianRandomWalk. As its name suggests GaussianRandomWalk is a vector valued distribution where the values of the vector form a random normal walk of length n, as specified by the shape argument. The scale of the innovations of the random walk, sigma, is specified in terms of the precision of the normally distributed innovations and can be a scalar or vector.
End of explanation
from pymc3 import variational
import scipy
with sp500_model:
    mu, sds, elbo = variational.advi(n=100000)
    step = NUTS(scaling=sp500_model.dict_to_array(sds)**2, is_cov=True)
    trace = sample(2000, step, start=mu, progressbar=True)
Explanation: Notice that we transform the log volatility process s into the volatility process by exp(-2*s). Here, exp is a Theano function, rather than the corresponding function in NumPy; Theano provides a large subset of the mathematical functions that NumPy does.
Also note that we have declared the Model name sp500_model in the first occurrence of the context manager, rather than splitting it into two lines, as we did for the first example.
Fitting
Before we draw samples from the posterior, it is prudent to find a decent starting value by finding a point of relatively high probability. For this model, the full maximum a posteriori (MAP) point over all variables is degenerate and has infinite density. But, if we fix log_sigma and nu it is no longer degenerate, so we find the MAP with respect only to the volatility process s keeping log_sigma and nu constant at their default values (remember that we set testval=.1 for sigma). We use the Limited-memory BFGS (L-BFGS) optimizer, which is provided by the scipy.optimize package, as it is more efficient for high dimensional functions and we have 400 stochastic random variables (mostly from s).
To achieve good convergence with NUTS, it is critical to find a good scaling. The MAP as found by find_MAP() is in our experience a poor choice. Instead, we run automatic differentiation variational inference (ADVI) to get a variational estimate of the posterior mean as well as the posterior standard deviations. We can use these estimates to initialize NUTS to achieve faster sampling due to better scaling.
End of explanation
traceplot(trace[200:], [nu, sigma]);
Explanation: We can check our samples by looking at the traceplot for nu and sigma.
End of explanation
fig, ax = plt.subplots(figsize=(15, 8))
returns.plot(ax=ax)
ax.plot(returns.index, 1/np.exp(trace['s',::5].T), 'r', alpha=.03);
ax.set(title='volatility_process', xlabel='time', ylabel='volatility');
ax.legend(['S&P500', 'stochastic volatility process'])
Explanation: Finally we plot the distribution of volatility paths by plotting many of our sampled volatility paths on the same graph. Each is rendered partially transparent (via the alpha argument in Matplotlib's plot function) so the regions where many paths overlap are shaded more darkly.
End of explanation
disaster_data = np.ma.masked_values([4, 5, 4, 0, 1, 4, 3, 4, 0, 6, 3, 3, 4, 0, 2, 6,
                            3, 3, 5, 4, 5, 3, 1, 4, 4, 1, 5, 5, 3, 4, 2, 5,
                            2, 2, 3, 4, 2, 1, 3, -999, 2, 1, 1, 1, 1, 3, 0, 0,
                            1, 0, 1, 1, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 1, 1,
                            0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2,
                            3, 3, 1, -999, 2, 1, 1, 1, 1, 2, 4, 2, 0, 0, 1, 4,
                            0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1], value=-999)
year = np.arange(1851, 1962)
plt.plot(year, disaster_data, 'o', markersize=8);
plt.ylabel("Disaster count")
plt.xlabel("Year")
Explanation: As you can see, the model correctly infers the increase in volatility during the 2008 financial crash. Moreover, note that this model is quite complex because of its high dimensionality and dependency-structure in the random walk distribution. NUTS as implemented in PyMC3, however, correctly infers the posterior distribution with ease.
Case study 2: Coal mining disasters
Consider the following time series of recorded coal mining disasters in the UK from 1851 to 1962 (Jarrett, 1979). The number of disasters is thought to have been affected by changes in safety regulations during this period. Unfortunately, we also have pair of years with missing data, identified as missing by a NumPy MaskedArray using -999 as the marker value. 
Next we will build a model for this series and attempt to estimate when the change occurred. At the same time, we will see how to handle missing data, use multiple samplers and sample from discrete random variables.
End of explanation
from pymc3 import DiscreteUniform, Poisson
from pymc3.math import switch
with Model() as disaster_model:
    switchpoint = DiscreteUniform('switchpoint', lower=year.min(), upper=year.max(), testval=1900)
    # Priors for pre- and post-switch rates number of disasters
    early_rate = Exponential('early_rate', 1)
    late_rate = Exponential('late_rate', 1)
    # Allocate appropriate Poisson rates to years before and after current
    rate = switch(switchpoint >= year, early_rate, late_rate)
    disasters = Poisson('disasters', rate, observed=disaster_data)
Explanation: Occurrences of disasters in the time series is thought to follow a Poisson process with a large rate parameter in the early part of the time series, and from one with a smaller rate in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations.
In our model, 
$$ 
\begin{aligned}
  D_t &\sim \text{Pois}(r_t), r_t= \begin{cases} 
   l, & \text{if } t \lt s \
   e, & \text{if } t \ge s 
   \end{cases} \
  s &\sim \text{Unif}(t_l, t_h)\       
  e &\sim \text{exp}(1)\
  l &\sim \text{exp}(1)  
\end{aligned}
$$
 the parameters are defined as follows: 
   * $D_t$: The number of disasters in year $t$
   * $r_t$: The rate parameter of the Poisson distribution of disasters in year $t$.
   * $s$: The year in which the rate parameter changes (the switchpoint).
   * $e$: The rate parameter before the switchpoint $s$.
   * $l$: The rate parameter after the switchpoint $s$.
   * $t_l$, $t_h$: The lower and upper boundaries of year $t$.
This model is built much like our previous models. The major differences are the introduction of discrete variables with the Poisson and discrete-uniform priors and the novel form of the deterministic random variable rate.
End of explanation
from pymc3 import Metropolis 
with disaster_model:
    step1 = NUTS([early_rate, late_rate])
    
    # Use Metropolis for switchpoint, and missing values since it accommodates discrete variables
    step2 = Metropolis([switchpoint, disasters.missing_values[0]])
    trace = sample(10000, step=[step1, step2])
Explanation: The logic for the rate random variable,
python
rate = switch(switchpoint &gt;= year, early_rate, late_rate)
is implemented using switch, a Theano function that works like an if statement. It uses the first argument to switch between the next two arguments.
Missing values are handled transparently by passing a MaskedArray or a pandas.DataFrame with NaN values to the observed argument when creating an observed stochastic random variable. Behind the scenes, another random variable, disasters.missing_values is created to model the missing values. All we need to do to handle the missing values is ensure we sample this random variable as well.
Unfortunately because they are discrete variables and thus have no meaningful gradient, we cannot use NUTS for sampling switchpoint or the missing disaster observations. Instead, we will sample using a Metroplis step method, which implements adaptive Metropolis-Hastings, because it is designed to handle discrete values.
We sample with both samplers at once by passing them to the sample function in a list. Each new sample is generated by first applying step1 then step2.
End of explanation
traceplot(trace);
Explanation: In the trace plot below we can see that there's about a 10 year span that's plausible for a significant change in safety, but a 5 year span that contains most of the probability mass. The distribution is jagged because of the jumpy relationship between the year switchpoint and the likelihood  and not due to sampling error.
End of explanation
import theano.tensor as T 
from theano.compile.ops import as_op
@as_op(itypes=[T.lscalar], otypes=[T.lscalar])
def crazy_modulo3(value):
    if value > 0: 
        return value % 3
    else :
        return (-value + 1) % 3
    
with Model() as model_deterministic:
    a = Poisson('a', 1)
    b = crazy_modulo3(a)
Explanation: Arbitrary deterministics
Due to its reliance on Theano, PyMC3 provides many mathematical functions and operators for transforming random variables into new random variables. However, the library of functions in Theano is not exhaustive, therefore Theano and PyMC3 provide functionality for creating arbitrary Theano functions in pure Python, and including these functions in PyMC models. This is supported with the as_op function decorator.
Theano needs to know the types of the inputs and outputs of a function, which are specified for as_op by itypes for inputs and otypes for outputs. The Theano documentation includes an overview of the available types.
End of explanation
from pymc3.distributions import Continuous
class Beta(Continuous):
    def __init__(self, mu, *args, **kwargs):
        super(Beta, self).__init__(*args, **kwargs)
        self.mu = mu
        self.mode = mu
    def logp(self, value):
        mu = self.mu
        return beta_logp(value - mu)
    
@as_op(itypes=[T.dscalar], otypes=[T.dscalar])
def beta_logp(value):
    return -1.5 * np.log(1 + (value)**2)
with Model() as model:
    beta = Beta('slope', mu=0, testval=0)
Explanation: An important drawback of this approach is that it is not possible for theano to inspect these functions in order to compute the gradient required for the Hamiltonian-based samplers. Therefore, it is not possible to use the HMC or NUTS samplers for a model that uses such an operator. However, it is possible to add a gradient if we inherit from theano.Op instead of using as_op. The PyMC example set includes a more elaborate example of the usage of as_op.
Arbitrary distributions
Similarly, the library of statistical distributions in PyMC3 is not exhaustive, but PyMC allows for the creation of user-defined functions for an arbitrary probability distribution. For simple statistical distributions, the DensityDist function takes as an argument any function that calculates a log-probability $log(p(x))$. This function may employ other random variables in its calculation. Here is an example inspired by a blog post by Jake Vanderplas on which priors to use for a linear regression (Vanderplas, 2014). 
```python
import theano.tensor as T
from pymc3 import DensityDist, Uniform
with Model() as model:
    alpha = Uniform('intercept', -100, 100)
# Create custom densities
beta = DensityDist('beta', lambda value: -1.5 * T.log(1 + value**2), testval=0)
eps = DensityDist('eps', lambda value: -T.log(T.abs_(value)), testval=1)
# Create likelihood
like = Normal('y_est', mu=alpha + beta * X, sd=eps, observed=Y)
```
For more complex distributions, one can create a subclass of Continuous or Discrete and provide the custom logp function, as required. This is how the built-in distributions in PyMC are specified. As an example, fields like psychology and astrophysics have complex likelihood functions for a particular process that may require numerical approximation. In these cases, it is impossible to write the function in terms of predefined theano operators and we must use a custom theano operator using as_op or inheriting from theano.Op. 
Implementing the beta variable above as a Continuous subclass is shown below, along with a sub-function using the as_op decorator, though this is not strictly necessary.
End of explanation
# Convert X and Y to a pandas DataFrame
import pandas 
df = pandas.DataFrame({'x1': X1, 'x2': X2, 'y': Y})
Explanation: Generalized Linear Models
Generalized Linear Models (GLMs) are a class of flexible models that are widely used to estimate regression relationships between a single outcome variable and one or multiple predictors. Because these models are so common, PyMC3 offers a glm submodule that allows flexible creation of various GLMs with an intuitive R-like syntax that is implemented via the patsy module.
The glm submodule requires data to be included as a pandas DataFrame. Hence, for our linear regression example:
End of explanation
from pymc3.glm import glm
with Model() as model_glm:
    glm('y ~ x1 + x2', df)
    trace = sample(5000)
Explanation: The model can then be very concisely specified in one line of code.
End of explanation
from pymc3.glm.families import Binomial
df_logistic = pandas.DataFrame({'x1': X1, 'y': Y > np.median(Y)})
with Model() as model_glm_logistic:
    glm('y ~ x1', df_logistic, family=Binomial())
Explanation: The error distribution, if not specified via the family argument, is assumed to be normal. In the case of logistic regression, this can be modified by passing in a Binomial family object.
End of explanation
from pymc3.backends import SQLite
with Model() as model_glm_logistic:
    glm('y ~ x1', df_logistic, family=Binomial())
    
    backend = SQLite('trace.sqlite')
    start = find_MAP()
    step = NUTS(scaling=start)
    trace = sample(5000, step=step, start=start, trace=backend)
summary(trace, varnames=['x1'])
Explanation: Backends
PyMC3 has support for different ways to store samples during and after sampling, called backends, including in-memory (default), text file, and SQLite. These can be found in pymc.backends:
By default, an in-memory ndarray is used but if the samples would get too large to be held in memory we could use the sqlite backend:
End of explanation
from pymc3.backends.sqlite import load
with basic_model:
    trace_loaded = load('trace.sqlite')
Explanation: The stored trace can then later be loaded using the load command:
End of explanation
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