- Introduction
- Installation
- Tutorial 1, linear regression
- Tutorial 2, linear regression with more than one predictor
- Unit tests
MathematicaStan is a package to interact with CmdStan from Mathematica.
It is developed under Linux and is compatible with Mathematica v11+
It should work under MacOS and also under Windows.
Author & contact: picaud.vincent at gmail.com
New MathematicaStan version 2.2!
Package test with last CmdStan v2.35.0, Mathematica 11.2, Linux
- Add some screenshots to the install procedure section
- CmdStan syntax changes have been included :
old current <- = increment_log_prob(…) target += … int<lower=0,upper=1> y[N]; array[N] int<lower=0, upper=1> y; - Check that unit tests and examples work.
New MathematicaStan version 2.1!
This version has been fixed and should now run under Windows.
I would like to thank Ali Ghaderi who had the patience to help me to debug the Windows version (I do not have access to this OS). Nothing would have been possible without him. All possibly remaining bugs are mine.
As a remainder also note that one should not use path/filename with
spaces (Make really does not like that). This consign is also true
under Linux or MacOS. See SO:can-gnu-make-handle-filenames-with-spaces
by example.
New MathematicaStan version 2.0!
This version uses Mathematica v11 and has been completely refactored
Caveat: breaking changes!
Note: the “old” MathematicaStan version based on Mathematica v8.0 is now archived in the v1 git branch.
First you must install CmdStan. Once this is done you get a directory containing stuff like:
bin doc examples Jenkinsfile LICENSE make makefile README.md runCmdStanTests.py src stan test-all.sh
With my configuration CmdStan is installed in:
~/ExternalSoftware/cmdstan-2.35.0
For Windows users it is possibly something like:
C:\\Users\\USER_NAME\\Documents\\R\\cmdstan-?.??.?
To install the Mathematica CmdStan package:
- open the
CmdStan.mfile with Mathematica. - install it using the Mathematica Notebook File->Install menu.
Fill in the pop-up windows as follows:
The first time the package is imported
<<CmdStan`you will get an error message:
CmdStan::cmdStanDirectoryNotDefined: CmdStan directory does not exist, use SetCmdStanDirectory[dir] to define it (with something like SetCmdStanDirectory["~/ExternalSoftware/cmdstan-2.35.0"])
This is normal as we must define the Stan StanCmd shell interface root directory.
With my configuration this is:
SetCmdStanDirectory["~/ExternalSoftware/cmdstan-2.35.0"]
For Windows user this is certainly something like:
SetCmdStanDirectory["C:\\Users\\USER_NAME\\Documents\\R\\cmdstan-?.??.?"]
Note: this location is recorded in the $CmdStanConfigurationFile file
and you will not have to redefine it every time you import the
CmdStan package.
You can use the file tutorial.wls or manually follow the instruction
below.
Import the package as usual
<<CmdStan`This package defines these functions (and symbols):
?CmdStan`*| CmdStan | GetStanOption | RemoveStanOption | StanOptionExistsQ | StanResultReducedKeys |
| CompileStanCode | GetStanResult | RunStan | StanOptions | StanResultReducedMetaKeys |
| ExportStanCode | GetStanResultMeta | SampleDefaultOptions | StanResult | StanVerbose |
| ExportStanData | ImportStanResult | SetCmdStanDirectory | StanResultKeys | VariationalDefaultOptions |
| GetCmdStanDirectory | OptimizeDefaultOptions | SetStanOption | StanResultMetaKeys | $CmdStanConfigurationFile |
For this tutorial we use a simple linear regression example and we will work in a temporary location:
SetDirectory[$TemporaryDirectory]/tmp
Define the Stan code
stanCode = "data
{
int<lower = 0> N;
vector[N] x;
vector[N] y;
}
parameters
{
real alpha;
real beta;
real<lower = 0> sigma;
}
model {
y ~normal(alpha + beta * x, sigma);
}";and export it
stanCodeFile = ExportStanCode["linear_regression.stan", stanCode]/tmp/linear_regression.stan
Stan code compilation is performed by
stanExeFile = CompileStanCode[stanCodeFile] (* Attention: this takes some time *)With my configuration I get
make: Entering directory '/home/picaud/ExternalSoftware/cmdstan-2.35.0' --- Translating Stan model to C++ code --- bin/stanc --o=/tmp/linear_regression.hpp /tmp/linear_regression.stan Model name=linear_regression_model Input file=/tmp/linear_regression.stan Output file=/tmp/linear_regression.hpp g++ -std=c++1y -pthread -Wno-sign-compare -O3 -I src -I stan/src -I stan/lib/stan_math/ -I stan/lib/stan_math/lib/eigen_3.3.3 -I stan/lib/stan_math/lib/boost_1.69.0 -I stan/lib/stan_math/lib/sundials_4.1.0/include -DBOOST_RESULT_OF_USE_TR1 -DBOOST_NO_DECLTYPE -DBOOST_DISABLE_ASSERTS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -c -MT /tmp/linear_regression.o -MT /tmp/linear_regression -include /tmp/linear_regression.hpp -include src/cmdstan/main.cpp -MM -E -MG -MP -MF /tmp/linear_regression.d /tmp/linear_regression.hpp --- Linking C++ model --- g++ -std=c++1y -pthread -Wno-sign-compare -O3 -I src -I stan/src -I stan/lib/stan_math/ -I stan/lib/stan_math/lib/eigen_3.3.3 -I stan/lib/stan_math/lib/boost_1.69.0 -I stan/lib/stan_math/lib/sundials_4.1.0/include -DBOOST_RESULT_OF_USE_TR1 -DBOOST_NO_DECLTYPE -DBOOST_DISABLE_ASSERTS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -include /tmp/linear_regression.hpp src/cmdstan/main.cpp stan/lib/stan_math/lib/sundials_4.1.0/lib/libsundials_nvecserial.a stan/lib/stan_math/lib/sundials_4.1.0/lib/libsundials_cvodes.a stan/lib/stan_math/lib/sundials_4.1.0/lib/libsundials_idas.a -o /tmp/linear_regression make: Leaving directory '/home/picaud/ExternalSoftware/cmdstan-2.35.0'
Note: if you do not want to have information printed you can use the StanVerbose option:
stanExeFile = CompileStanCode[stanCodeFile, StanVerbose -> False]Let’s simulate some data:
σ = 3; α = 1; β = 2;
n = 20;
X = Range[n];
Y = α + β*X + RandomVariate[NormalDistribution[0, σ], n];
Show[Plot[α + β*x, {x, Min[X], Max[X]}],
ListPlot[Transpose@{X, Y}, PlotStyle -> Red]]
The data are stored in a Association and then exported thanks to the
ExportStanData function.
stanData = <|"N" -> n, "x" -> X, "y" -> Y|>;
stanDataFile = ExportStanData[stanExeFile, stanData]/tmp/linear_regression.data.R
Note: this function returns the created file
name /tmp/linear_regression.data.R. Its first argument, stanExeFile
is simply the Stan executable file name with its path. The
ExportStanData[] function modifies the file name extension and
replace it with “.data.R”, but you can use it with
any file name:
ExportStanData["my_custom_path/my_custom_filename.data.R",stanData]We are now able to run the stanExeFile executable.
Let’s start by maximizing the likelihood
stanResultFile = RunStan[stanExeFile, OptimizeDefaultOptions]Running: /tmp/linear_regression method=optimize data file=/tmp/linear_regression.data.R output file=/tmp/linear_regression.csv
method = optimize
optimize
algorithm = lbfgs (Default)
lbfgs
init_alpha = 0.001 (Default)
tol_obj = 9.9999999999999998e-13 (Default)
tol_rel_obj = 10000 (Default)
tol_grad = 1e-08 (Default)
tol_rel_grad = 10000000 (Default)
tol_param = 1e-08 (Default)
history_size = 5 (Default)
iter = 2000 (Default)
save_iterations = 0 (Default)
id = 0 (Default)
data
file = /tmp/linear_regression.data.R
init = 2 (Default)
random
seed = 2775739062
output
file = /tmp/linear_regression.csv
diagnostic_file = (Default)
refresh = 100 (Default)
Initial log joint probability = -8459.75
Iter log prob ||dx|| ||grad|| alpha alpha0 # evals Notes
19 -32.5116 0.00318011 0.00121546 0.9563 0.9563 52
Optimization terminated normally:
Convergence detected: relative gradient magnitude is below tolerance
The stanResultFile variable contains now the csv result file:
/tmp/linear_regression.csv
Note: again, if you do not want to have printed output, use the StanVerbose->False option.
stanResultFile = RunStan[stanExeFile, OptimizeDefaultOptions,StanVerbose->False]Note: the method we use is defined by the second argument
OptimizeDefaultOptions. If you want to use Variational Bayes or HMC
sampling you must use
RunStan[stanExeFile, VariationalDefaultOptions]or
RunStan[stanExeFile, SampleDefaultOptions]Note: option management will be detailed later in this tutorial.
To load CSV result file, do
stanResult = ImportStanResult[stanResultFile]which prints
file: /tmp/linear_regression.csv
meta: lp__
parameter: alpha , beta , sigma
To access estimated variable α, β and σ, simply do:
GetStanResultMeta[stanResult, "lp__"]
αe=GetStanResult[stanResult, "alpha"]
βe=GetStanResult[stanResult, "beta"]
σe=GetStanResult[stanResult, "sigma"]which prints:
{-32.5116}
{2.51749}
{1.83654}
{3.08191}
Note: as with likelihood maximization we only have a point estimation, the returned values are lists of one number.
You can plot the estimated line:
Show[Plot[{αe + βe*x, α + β*x}, {x, Min[X],Max[X]}, PlotLegends -> "Expressions"],
ListPlot[Transpose@{X, Y}, PlotStyle -> Red]]
We want to solve the same problem but using variational inference.
As explained before we must use
stanResultFile = RunStan[stanExeFile, VariationalDefaultOptions]instead of
stanResultFile = RunStan[stanExeFile, OptimizeDefaultOptions]However, please note that running this command will erase
stanResultFile which is the file where result are exported. To avoid
this we can modify the output file name by modifying option values.
The default option values are stored in the write-protected
VariationalDefaultOptions variable.
To modify them we must first copy this protected symbol:
myOpt=VariationalDefaultOptionswhich prints
method=variational
The option values are printed when you run the RunStan command:
method = variational
variational
algorithm = meanfield (Default)
meanfield
iter = 10000 (Default)
grad_samples = 1 (Default)
elbo_samples = 100 (Default)
eta = 1 (Default)
adapt
engaged = 1 (Default)
iter = 50 (Default)
tol_rel_obj = 0.01 (Default)
eval_elbo = 100 (Default)
output_samples = 1000 (Default)
id = 0 (Default)
data
file = (Default)
init = 2 (Default)
random
seed = 2784129612
output
file = output.csv (Default)
diagnostic_file = (Default)
refresh = 100 (Default)
We have to modify the output file option value. This can be done by:
myOpt = SetStanOption[myOpt, "output.file", FileNameJoin[{Directory[], "myOutputFile.csv"}]]which prints:
method=variational output file=/tmp/myOutputFile.csv
Now we can run Stan:
myOutputFile=RunStan[stanExeFile, myOpt, StanVerbose -> False]which must print:
/tmp/myOutputFile.csv
Now import this CSV file:
myResult = ImportStanResult[myOutputFile]which prints:
file: /tmp/myOutputFile.csv
meta: lp__ , log_p__ , log_g__
parameter: alpha , beta , sigma
As before you can use:
GetStanResult[myResult,"alpha"]to get alpha parameter value, but now you will get a list of 1000 sample:
{2.03816, 0.90637, ..., ..., 1.22068, 1.66392}
Instead of the full sample list we are often interested by sample mean, variance… You can get these quantities as follows:
GetStanResult[Mean, myResult, "alpha"]
GetStanResult[Variance, myResult, "alpha"]which prints:
2.0353 0.317084
You can also get the sample hstogram as simply as:
GetStanResult[Histogram, myResult, "alpha"]
We provide further details concerning option related functions.
To recap the first step is to perform a copy of the write-protected default option values. By example to modify default MCMC option values the first step is:
myOpt = SampleDefaultOptionsThe available option are:
method = sample (Default)
sample
num_samples = 1000 (Default)
num_warmup = 1000 (Default)
save_warmup = 0 (Default)
thin = 1 (Default)
adapt
engaged = 1 (Default)
gamma = 0.050000000000000003 (Default)
delta = 0.80000000000000004 (Default)
kappa = 0.75 (Default)
t0 = 10 (Default)
init_buffer = 75 (Default)
term_buffer = 50 (Default)
window = 25 (Default)
algorithm = hmc (Default)
hmc
engine = nuts (Default)
nuts
max_depth = 10 (Default)
metric = diag_e (Default)
metric_file = (Default)
stepsize = 1 (Default)
stepsize_jitter = 0 (Default)
id = 0 (Default)
data
file = /tmp/linear_regression.data.R
init = 2 (Default)
random
seed = 3714706817 (Default)
output
file = /tmp/linear_regression.csv
diagnostic_file = (Default)
refresh = 100 (Default)
sig_figs = -1 (Default)
If we want to modify:
method = sample (Default)
sample
num_samples = 1000 (Default)
num_warmup = 1000 (Default)
and
method = sample (Default)
sample
algorithm = hmc (Default)
hmc
engine = nuts (Default)
nuts
max_depth = 10 (Default)
you must proceed as follows. For each hierarchy level use a “.” as separator and do not forget to rewrite “=” with the associated value. With our example this gives:
myOpt = SetStanOption[myOpt, "adapt.num_samples", 2000]
myOpt = SetStanOption[myOpt, "adapt.num_warmup", 1500]
myOpt = SetStanOption[myOpt, "algorithm=hmc.engine=nuts.max_depth", 5]Now you can run the sampler with these new option values:
stanResultFile = RunStan[stanExeFile, myOpt]which should print:
method = sample (Default)
sample
num_samples = 2000
num_warmup = 1500
save_warmup = 0 (Default)
thin = 1 (Default)
adapt
engaged = 1 (Default)
gamma = 0.050000000000000003 (Default)
delta = 0.80000000000000004 (Default)
kappa = 0.75 (Default)
t0 = 10 (Default)
init_buffer = 75 (Default)
term_buffer = 50 (Default)
window = 25 (Default)
algorithm = hmc (Default)
hmc
engine = nuts (Default)
nuts
max_depth = 5
metric = diag_e (Default)
metric_file = (Default)
stepsize = 1 (Default)
stepsize_jitter = 0 (Default)
id = 0 (Default)
data
file = /tmp/linear_regression.data.R
init = 2 (Default)
random
seed = 3720771451 (Default)
output
file = /tmp/linear_regression.csv
diagnostic_file = (Default)
refresh = 100 (Default)
sig_figs = -1 (Default)
stanc_version = stanc3 b25c0b64
stancflags =
Gradient evaluation took 1.3e-05 seconds
1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
Adjust your expectations accordingly!
Iteration: 1 / 3500 [ 0%] (Warmup)
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Elapsed Time: 0.053 seconds (Warm-up)
0.094 seconds (Sampling)
0.147 seconds (Total)
You can check than the new option values have been taken into account:
num_samples = 2000
num_warmup = 1500
algorithm = hmc (Default)
hmc
engine = nuts (Default)
nuts
max_depth = 5
You can get back the modified values as follows:
GetStanOption[myOpt, "adapt.num_warmup"]
GetStanOption[myOpt, "algorithm=hmc.engine=nuts.max_depth"]which prints
1500 5
Caveat: if the option was not defined (by SetStanOption) the function
returns $Failed.
To erase an option value (and use its default value) use:
myOpt = RemoveStanOption[myOpt, "algorithm=hmc.engine=nuts.max_depth"]which prints
method=sample adapt num_samples=2000 num_warmup=1500
By now the parameters alpha, beta, sigma, were scalars. We will see how to handle parameters that are vectors or matrices.
We use second section of the linear regression example, entitled “Matrix notation and Vectorization”.
The β parameter is now a vector of size K.
stanCode = "data {
int<lower=0> N; // number of data items
int<lower=0> K; // number of predictors
matrix[N, K] x; // predictor matrix
vector[N] y; // outcome vector
}
parameters {
real alpha; // intercept
vector[K] beta; // coefficients for predictors
real<lower=0> sigma; // error scale
}
model {
y ~ normal(x * beta + alpha, sigma); // likelihood
}";
stanCodeFile = ExportStanCode["linear_regression_vect.stan", stanCode];
stanExeFile = CompileStanCode[stanCodeFile];Here we use {x,x²,x³} as predictors, with their coefficients β = {2,0.1,0.01} so that the model is
y = α + β1 x + β2 x² + β3 x³ + ε
where ε follows a normal distribution.
σ = 3; α = 1; β1 = 2; β2 = 0.1; β3 = 0.01;
n = 20;
X = Range[n];
Y = α + β1*X + β2*X^2 + β3*X^3 + RandomVariate[NormalDistribution[0, σ], n];
Show[Plot[α + β1*x + β2*x^2 + β3*x^3, {x, Min[X], Max[X]}],
ListPlot[Transpose@{X, Y}, PlotStyle -> Red]]
The expression
y = α + β1 x + β2 x² + β3 x³ + ε
is convenient for random variable manipulations. However in practical computations where we have to evaluate:
y[i] = α + β1 x[i] + β2 (x[i])² + β3 (x[i])³ + ε[i], for i = 1..N
it is more convenient to rewrite this in a “vectorized form”:
y = α + X.β + ε
where X is a KxN matrix of columns X[:,j] = j th-predictor = (x[:])^j and α a vector of size N with constant components = α.
Thus data is exported as follows:
stanData = <|"N" -> n, "K" -> 3, "x" -> Transpose[{X,X^2,X^3}], "y" -> Y|>;
stanDataFile = ExportStanData[stanExeFile, stanData]Note: as Mathematica stores its matrices rows by rows (the C
language convention) we have to transpose {X,X^2,X^3} to get the
right matrix X.
We can now run Stan using the Hamiltonian Monte Carlo (HMC) method:
stanResultFile = RunStan[stanExeFile, SampleDefaultOptions]which prints:
Running: /tmp/linear_regression_vect method=sample data file=/tmp/linear_regression_vect.data.R output file=/tmp/linear_regression_vect.csv
method = sample (Default)
sample
num_samples = 1000 (Default)
num_warmup = 1000 (Default)
save_warmup = 0 (Default)
thin = 1 (Default)
adapt
engaged = 1 (Default)
gamma = 0.050000000000000003 (Default)
delta = 0.80000000000000004 (Default)
kappa = 0.75 (Default)
t0 = 10 (Default)
init_buffer = 75 (Default)
term_buffer = 50 (Default)
window = 25 (Default)
algorithm = hmc (Default)
hmc
engine = nuts (Default)
nuts
max_depth = 10 (Default)
metric = diag_e (Default)
metric_file = (Default)
stepsize = 1 (Default)
stepsize_jitter = 0 (Default)
id = 0 (Default)
data
file = /tmp/linear_regression_vect.data.R
init = 2 (Default)
random
seed = 3043713420
output
file = /tmp/linear_regression_vect.csv
diagnostic_file = (Default)
refresh = 100 (Default)
Gradient evaluation took 4e-05 seconds
1000 transitions using 10 leapfrog steps per transition would take 0.4 seconds.
Adjust your expectations accordingly!
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Elapsed Time: 0.740037 seconds (Warm-up)
0.60785 seconds (Sampling)
1.34789 seconds (Total)
As before,
stanResult = ImportStanResult[stanResultFile]load the generated CSV file and prints:
file: /tmp/linear_regression_vect.csv
meta: lp__ , accept_stat__ , stepsize__ , treedepth__ , n_leapfrog__ , divergent__ , energy__
parameter: alpha , beta 3, sigma
Compared to the scalar case, the important thing to notice is the beta 3. That means that β is not a scalar anymore but a vector of size 3
Note: here β is a vector, but if it had been a 3x5 matrix we would
have had β 3x5 printed instead.
A call to
GetStanResult[stanResult, "beta"]returns a vector of size 3 but where each component is a list of 1000 sample (for β1, β2 and β3).
As before it generally useful to summarize this sample with function like mean or histogram:
GetStanResult[Mean, stanResult, "beta"]
GetStanResult[Histogram, stanResult, "beta"]prints:
{3.30321, -0.010088, 0.0126913}
and plots:
This is the moment to digress about Keys. If you try:
StanResultKeys[stanResult]
StanResultMetaKeys[stanResult]this will print:
{"alpha", "beta.1", "beta.2", "beta.3", "sigma"}
{"lp__", "accept_stat__", "stepsize__", "treedepth__", "n_leapfrog__", "divergent__", "energy__"}
These functions are useful to get the complete list of keys. Note
that, as β is an 1D-array of size 1 we have beta.1, beta.2, beta.3. If
β was a NxM matrix, the list of keys would have been: beta.1.1,
beta.1.2,... beta.N.M.
There is also reduced keys functions:
StanResultReducedKeys[stanResult]
StanResultReducedMetaKeys[stanResult]which print
{"alpha", "beta", "sigma"}
{"lp__", "accept_stat__", "stepsize__", "treedepth__", "n_leapfrog__", "divergent__", "energy__"}
As you can see the reduced keys functions collect and discard indices to keys associated to arrays.
When accessing a parameter you can work at the component level or globally:
GetStanResult[Mean, stanResult, "beta.2"]
GetStanResult[Mean, stanResult, "beta"]which prints
-0.010088
{3.30321, -0.010088, 0.0126913}
You can run tests/CmdStan_test.wl to check that everything works as expected.