A tutorial by Dan Foreman-Mackey (CCA/Flatiron)
In this tutorial, we will go through the implementation of two MCMC problems using three popular Python packages for probabilistic inference: emcee, dynesty, and PyMC3. One of the goals of this worksheet is to demonstrate the similarities and (expecially) differences between the usage and performance of the different packages. These are all great libraries with different benefits and shortcomings and I'm hoping that, by the end of this tutorial, you will know some of the things that you need to know to choose the right tool for your specific project. There are many other tools (even many others written in Python) that can be used for probabilistic inference, but I've chosen these three because they approximately span the range of general purpose tools that are commonly used in astro and they're the ones that I'm most familiar with.
Broadly speaking, the three packages can be roughly described as follows:
emcee is a black box MCMC routine (it doesn't need your code to be written in a specific way) that doesn't require much tuning. It is going to be best suited to low dimensional problems (less than about 10 parameters) with a large dynamic range or significant covariances between the parameters. It doesn't scale well to high dimensional problems and multi-modal posterior densities can be seriously problematic. Note: For this tutorial, we'll be using a pre-release version emcee: 3.0. To install that version, run pip install emcee==3.0rc2.
dynesty is a modern implementation of nested sampling that works well for relatively low dimensional problems even if there are multiple modes in the posterior. Nested sampling is an algorithm for computing the "marginalized likelihood" or "evidence integral", but it also produces posterior samples (much like those produced by MCMC) as a byproduct. This is the feature that we'll be using today. dynesty and emcee have similar user interfaces so it shouldn't be too hard to switch between these two samplers in most cases.
PyMC3 is both a model building language and an MCMC engine. This can make the learning curve might be a bit steeper, but the initial legwork can be worth it in many cases. The main benefit of PyMC3 compared to the other libraries is that it scales much more efficiently to high dimensional problems. In many cases, PyMC3 can efficiently sample models with hundreds or thousands of parameters where the other samplers would really struggle. The biggest restriction when using PyMC3 is that you'll need to write your model using PyMC3's specific model building language. That means that you won't be able to easily use that Fortran code that your advisor's advisor wrote to evaluate your model. Sometimes this doesn't matter, but sometimes it does!
Below, we will go through two examples that demonstrate the usage of these libraries.
The first example is one that you'll see in one form or another about a million times this week: fitting a line to data. This is a relatively simple problem that will help give you some intuition about the relationships between the different libraries. There is also a challenge problem that will help give some intuition for how you would go about generalizing these implementations.
The second example is a more realistic problem (fitting radial velocity exoplanets) that will illustrate some real world issues that you'll run into when fitting real data, and give some ideas for how to deal with them.
To start, let's set up our environment (so that the plots look a little nicer) and make sure that we can import all of the modules that we're going to need.
In [1]:
%matplotlib inline
In [2]:
# There are a bunch of functions useful for this tutorial implemented in the
# advanced_sampling_utils.py file that should be in the same directory as
# this notebook.
from advanced_sampling_utils import setup_plotting
setup_plotting()
In [3]:
# Basic scientific Python modules
import numpy as np
import scipy.stats
import matplotlib.pyplot as plt
# For plotting: https://corner.readthedocs.io
import corner
# Sampling modules:
import emcee
import dynesty
import pymc3 as pm
# This is needed for our second pymc3 example below
import theano.tensor as tt
If any errors appeared, let me know and we'll work it out. If not, we're good to go!
In this example, we're going to fit a line to data, but we'll make it very slightly non-trivial by fitting for the amplitude of the noise as well as the slope and intercept of the line. This problem has been written about all over the place, and you'll hear a few other discussions of it this week, but we'll start with a quick summary of the model so that we have consistent notation.
The model that we'll fit has three parameters: the slope $m$, the intercept $b$, and the logarithm of the size of errorbar $\log \sigma$. Note: here and thoughout this tutorial, $\log$ means the natural logarithm, not $\log_{10}$. For some setting of these parameters, let's simulate some fake data:
In [4]:
np.random.seed(502489)
m_true = 3.1
b_true = -0.01
logs_true = np.log(3.5)
x = np.random.uniform(-5, 5, 25)
y = m_true * x + b_true + np.exp(logs_true)*np.random.randn(len(x))
plt.plot(x, y, ".k")
plt.xlabel("x")
plt.ylabel("y");
Instead of fitting these parameters directly, following the advice from this blogpost, we'll fit using the angle between the line and a horizontal line $\theta = \arctan m$, and the "perpendicular intercept" $b_\perp = b\,\cos\theta$.
After specifying the parameterization, we must then specify the prior and likelihood functions (or, more specifically, the logarithm of these functions). We're going to choose a separable prior:
$$ p(b_\perp,\,\theta,\,\log\sigma) = p(b_\perp)\,p(\theta)\,p(\log \sigma) $$and we will use uniform priors for each of these parameters:
$$ \log p(b_\perp) = \left\{\begin{array}{ll}\log(1/20) & \mathrm{if}\,-10 < b_\perp < 10 \\ -\infty & \mathrm{otherwise}\end{array}\right. $$$$ \log p(\theta) = \left\{\begin{array}{ll}\log(1/\pi) & \mathrm{if}\,-\pi/2 < \theta < \pi/2 \\ -\infty & \mathrm{otherwise}\end{array}\right. $$$$ \log p(\log \sigma) = \left\{\begin{array}{ll}\log(1/20) & \mathrm{if}\,-10 < \log \sigma < 10 \\ -\infty & \mathrm{otherwise}\end{array}\right. $$Then, we will assume Gaussian uncertainties with standard deviation $\sigma$ for the observation model. In this case, the (log-) likelihood function is:
$$ \log p(\{y_n\}\,|\,\{x_n\},\,b_\perp,\,\theta,\,\log\sigma) = -\frac{1}{2}\sum_{n=1}^N \left[\frac{(y_n - m\,x_n - b)^2}{\sigma^2} + \log(2\,\pi\,\sigma^2)\right] $$where $m$ and $b$ are computed as $m = \tan(\theta)$ and $b = b_\perp / \cos\theta$.
This is now enough information to start implementing the fits.
The only function that emcee requires to run MCMC is a function that takes a vector of parameters ($b_\perp$, $\theta$, and $\log\sigma$, in this case) and returns the log probability (evaluated up to a constant) for those parameters. However, it is often useful to separate the problem a little more than that. In the following cell, you should implement the following 4 functions:
emcee_get_params(params) that takes a vector of parameters (bperp, theta, and logs) and returns the parameters that the model needs: m, b, and logs.emcee_logprior(params) that takes a vector of parameters (bperp, theta, and logs) and returns the value of the log prior function (up to a constant, the normalization doesn't matter). This function should reject parameter sets that are outside of the allowed prior range by returning -np.inf.emcee_loglike(params) that takes a vector of parameters (bperp, theta, and logs) and returns the value of the log likelihood function (up to a constant).emcee_logprob(params) that takes a vector of parameters (bperp, theta, and logs) and returns the value of the log posterior probability (up to a constant). This should be implemented as the sum of the output from emcee_logprior and emcee_loglike, and I've actually already inserted an implementation that should work, but you might want to double check to make sure that it does the right thing when the parameters are outside of the prior bounds (return -np.inf if the log prior value is -np.inf).At the bottom of the cell, there is a set of unit tests that will check your implementation to make sure that your implementation is correct. If you see any errors below, try to fix the function that is failing the test.
In [5]:
def emcee_get_params(params):
#
# YOUR CODE GOES HERE
#
bperp, theta, logs = params
m = np.tan(theta)
b = bperp / np.cos(theta)
return m, b, logs
def emcee_logprior(params):
#
# YOUR CODE GOES HERE
#
bperp, theta, logs = params
if not -10 < bperp < 10:
return -np.inf
if not -0.5*np.pi < theta < 0.5*np.pi:
return -np.inf
if not -10 < logs < 10:
return -np.inf
return 0.0
def emcee_loglike(params):
#
# YOUR CODE GOES HERE
#
m, b, logs = emcee_get_params(params)
model = m * x + b
return -0.5 * np.sum((y - model)**2 / np.exp(2*logs) + 2*logs)
def emcee_logprob(params):
# I've actually implemented this function here for you
lp = emcee_logprior(params)
if not np.isfinite(lp):
return -np.inf
return lp + emcee_loglike(params)
# Now we run the unit tests to make sure that the functions are implemented correctly
from advanced_sampling_utils import test_emcee_functions
test_emcee_functions(emcee_get_params, emcee_logprior, emcee_loglike, emcee_logprob, x, y)
In the following cell, we'll go through the steps to initialize and run our sampling. This includes the following steps (and tuning parameter choices):
First, we need to initialize our sampler. For emcee, we need to select a point in parameter space for each "walker" in the ensemble. In this case, we'll use 100 walkers (I chose that number pretty arbitrarily), but you might find that you get better performance using a different choice. We'll initialize the walker coordinates by choosing a valid point in the parameter space and then add a small Gaussian random number to each walker coordinate so that they're not right on top of each other.
Next, we run our MCMC starting from the initial coordinates. For now, we'll just run 4000 steps, but you might want to use an automated convergence test when you use this in your research.
In [6]:
np.random.seed(9281)
init = np.array([0.0, 0.0, 0.0])
init = init + 0.1 * np.random.randn(100, 3)
emcee_sampler = emcee.EnsembleSampler(len(init), 3, emcee_logprob)
emcee_sampler.run_mcmc(init, 4000, progress=True);
After running an MCMC, it's always a good idea to run some convergence diagnostics.
With emcee, the recommended convergence diagnostic is the integrated autocorrelation time.
To get an estimate for the autocorrelation time for the chain for each parameter (Note: there isn't just one autocorrelation time per sampling, there's one for each scalar that you want to compute from the chain), we can use the get_autocorr_time method:
In [7]:
tau = emcee_sampler.get_autocorr_time()
print("Integrated autocorrelation times:")
for i, k in enumerate(["bperp", "theta", "logs"]):
print("{0}: {1}".format(k, tau[i]))
Next, let's discard approximately 5 autocorrelation times worth of samples as "burn-in" (the phase where the sampler "forgets" about its initialization) and then estimate the number of effective samples.
In [8]:
tau_max = int(tau.max())
emcee_samples = emcee_sampler.get_chain(discard=5*tau_max, flat=True)
emcee_samples = np.array(emcee_get_params(emcee_samples.T)).T
print("Effective number of samples for each parameter:")
print(len(emcee_samples) / emcee_sampler.get_autocorr_time(discard=5*tau_max))
That's a lot of samples! (Probably more than we need...)
Then it can also be useful to plot the corner plot for the sampling to visualize the covariances between the parameters. Since this fit was based on simulated data, we can also overplot the true values of the parameters; it looks like we recovered the truth!
In [9]:
corner.corner(emcee_samples, labels=["m", "b", "logs"],
truths=[m_true, b_true, logs_true]);
One last plot that can be good to look at is the posterior prediction of the model.
In this case, that would be a representation of the lines that are consistent with the data.
There are two different ways that I like to represent this: (a) as samples (using the plt.plot function with the alpha parameter) or (b) as a contour (using the plt.fill_between function).
Select one or two of these options and make a plot below with the data and the posterior lines overplotted.
If you choose to make a contour, a common choice would be to fill the area between the 16th and 84th percentiles of the posterior (check out the np.percentile function).
In [10]:
#
# YOUR CODE GOES HERE
#
x0 = np.linspace(-6, 6, 100)
lines = emcee_samples[:, 0] * x0[:, None] + emcee_samples[:, 1]
q = np.percentile(lines, [16, 50, 84], axis=1)
plt.fill_between(x0, q[0], q[2], color="C0", alpha=0.3)
plt.plot(x0, q[1], color="C0")
plt.xlim(-6, 6)
plt.plot(x, y, ".k")
plt.xlabel("x")
plt.ylabel("y");
Now let's update our model to use it with dynesty.
The only new function that we need to implement is a function called dynesty_prior_transform that takes samples on the unit cube and converts them to the parameter space that we're sampling.
The dynesty docs for this function are here.
This particular example is pretty simple, we need to convert 3 numbers in the range [0, 1] to three numbers in the ranges [-10, 10], [-pi/2, pi/2], and [-10, 10] respectively.
As above, implement your prior transform function below.
It should take a numpy array with three elements in the range [0, 1] and it should return a numpy array with entries (bperp, theta, logs).
Once your implementation is correct, the unit test should pass.
In [11]:
def dynesty_prior_transform(u):
#
# YOUR CODE GOES HERE
#
bperp = -10 + 20 * u[0]
theta = -0.5*np.pi + np.pi * u[1]
logs = -10 + 20 * u[2]
return np.array([bperp, theta, logs])
def dynesty_get_params(params):
# This function is the same as the one above for emcee...
return emcee_get_params(params)
def dynesty_loglike(params):
# So is this one!
return emcee_loglike(params)
# Now we run the unit tests to make sure that the functions are implemented correctly
from advanced_sampling_utils import test_dynesty_functions
test_dynesty_functions(dynesty_get_params, dynesty_prior_transform, dynesty_loglike, x, y)
Once all your tests pass, let's run the nested sampling algorithm.
Unlike emcee, dynesty doesn't require an initialization step. Instead, it will start by sampling from the prior.
Also unlike emcee, dynesty uses a convergence diagnostic as a stopping criterion so you won't choose the number of model evaluations that the sampler will perform.
Instead, you can tune the convergence criteria following the documentation here.
After the sampler converges, we re-sample the results using the resample_equal function in order to get effectively independent samples from the posterior density.
In [ ]:
# Run the sampler
np.random.seed(3028)
dynesty_sampler = dynesty.NestedSampler(dynesty_loglike, dynesty_prior_transform, 3)
dynesty_sampler.run_nested()
# Resample the results to get samples from the posterior
dynesty_results = dynesty_sampler.results
dynesty_samples = dynesty.utils.resample_equal(
dynesty_results.samples, np.exp(dynesty_results.logwt - dynesty_results.logz[-1])
)
dynesty_samples = np.array(dynesty_get_params(dynesty_samples.T)).T
print("Effective number of samples:")
print(np.any(np.abs(np.diff(np.sort(dynesty_samples, axis=0), axis=0)) > 1e-10, axis=1).sum())
In the following cells, make the two results plots that are equivalent for the ones above: the corner plot and the posterior predictions. Studying these plots, convince yourself that the results are the same and try to understand/explain any differences.
In [13]:
corner.corner(dynesty_samples, labels=["m", "b", "logs"],
truths=[m_true, b_true, logs_true]);
In [14]:
x0 = np.linspace(-6, 6, 100)
lines = dynesty_samples[:, 0] * x0[:, None] + dynesty_samples[:, 1]
q = np.percentile(lines, [16, 50, 84], axis=1)
plt.fill_between(x0, q[0], q[2], color="C0", alpha=0.3)
plt.plot(x0, q[1], color="C0")
plt.xlim(-6, 6)
plt.plot(x, y, ".k")
plt.xlabel("x")
plt.ylabel("y");
Finally, let's implement the same fit using PyMC3. The PyMC3 modeling interface is very different than the interface to emcee and dynesty, and I've found that the learning curve can be somewhat steeper, but it can be very powerful once you learn it!
One major difference is that you won't be implementing a function that evaluates the log probability.
Instead, you should define the distribution for each parameter and the relationships between them.
For example, since the $b_\perp$ (bperp) parameter should have a uniform prior between -10 and 10, we should write:
bperp = pm.Uniform("bperp", lower=-10, upper=10)
to define that variable.
Similarly, the likelihood will be defined as a distribution (a pm.Normal in this case) with the observed parameter set to the data - that's how we condition on the observations.
Another distribution that will come in handy is the pm.Deterministic distribution.
This let's us track the value of some parameters for each step in the chain without re-computing them afterwards.
For example, we want to track that value of the slope m for each step.
To do that, we would write:
m = pm.Deterministic("m", pm.math.tan(theta))
since $m = \tan\theta$.
You should note here that we didn't use np.tan.
That's because we cannot use numpy functions with pymc3 objects.
Instead, we must use the math functions defined in pm.math or theano.tensor which we imported above as tt.
pm.math has many of the mathematical functions that we need, but if something isn't included in that module, it's worth looking at the mathematical functions defined in theano.tensor to see if it's there.
In the cell below, implement the model using PyMC3.
It should have 3 parameters with uniform prior distributions: (1) bperp, (2) theta, and (3) logs.
It should also have two pm.Deterministic parameters for m and b calculated using pm.math or theano.tensor.
Finally, it should have a pm.Normal distribution with the observed parameter set.
Here are some links that might be useful references:
Once you have your model implemented, it should pass the unit test below.
In [15]:
# Your model needs to be *inside* this pm.Model "context manager":
with pm.Model() as pymc3_model:
bperp = pm.Uniform("bperp", lower=-10, upper=10)
#
# YOUR CODE GOES HERE
#
theta = pm.Uniform("theta", lower=-0.5*np.pi, upper=0.5*np.pi)
logs = pm.Uniform("logs", lower=-10, upper=10)
m = pm.Deterministic("m", pm.math.tan(theta))
b = pm.Deterministic("b", bperp / pm.math.cos(theta))
model = m * x + b
pm.Normal("loglike", mu=model, sd=pm.math.exp(logs), observed=y)
from advanced_sampling_utils import test_pymc3_model
test_pymc3_model(pymc3_model, x, y)
To sample using PyMC3, you will sometimes need to provide an initial guess for the parameters.
This is often best done using the testval argument in the distribution specifications (as you'll see below), but for this model, we can just sample starting from the default values.
So we'll just sample using the pm.sample function which will use the No U-Turn Sampling algorithm to generate samples from the posterior density.
We need to specify the number of samples to run as burn-in (called tune in PyMC3) where the parameters of the sampler are automatically adjusted to improve the performance of the sampling.
Then you'll also need to specify the number of samples to run for the final chain after burn-in completes (this parameter is called draws).
There are a lot of tuning parameters for this algorithm so you'll want to run more burn-in steps than you might expect from your previous experience.
As a useful starting point, I've found that setting tune and draws to the same value is generally about right.
In this case, we'll use 5000 steps for both tune and draws.
In [16]:
with pymc3_model:
np.random.seed(802941)
trace = pm.sample(tune=5000, draws=5000, chains=2)
As usual, after running our sampler, we should check some convergence diagnostics.
With PyMC3, we'll use the pm.summary function:
In [17]:
pm.summary(trace)
Out[17]:
For our purposes, you should mainly pay attention to the n_eff and Rhat columns.
The Rhat column is the $\hat{R}$ statistic from Brooks & Gelman (1998) and it should be close to 1 for a converged chain.
The n_eff column provides an estimate of the effective number of samples for each parameter.
In the cells below, make the two results plots like above (the pm.trace_to_dataframe function can be useful, but it's not necessary).
In [18]:
pymc3_samples = pm.trace_to_dataframe(trace, varnames=["m", "b", "logs"])
corner.corner(pymc3_samples, truths=[m_true, b_true, logs_true]);
In [19]:
x0 = np.linspace(-6, 6, 100)
lines = trace["m"] * x0[:, None] + trace["b"]
q = np.percentile(lines, [16, 50, 84], axis=1)
plt.fill_between(x0, q[0], q[2], color="C0", alpha=0.3)
plt.plot(x0, q[1], color="C0")
plt.xlim(-6, 6)
plt.plot(x, y, ".k")
plt.xlabel("x")
plt.ylabel("y");
A reasonable generalization of the above example is to include support for uncertainties on the $x$ coordinates. There has been quite a bit written on this topic - including a blog post that I wrote - and there is cleverness that you can use to simplify the problem. But in this problem, you should choose one of the three samplers above and generalize your implementation to include $N$ new parameters $\hat{x}_n$ (where $N$ is the number of datapoints) and $\hat{x}_n$ is the "true" value of $x_n$ that you're fitting. You can use any of the three sampling libraries, but this is a case where PyMC3 really shines (so that's what I'd suggest trying!).
Here's a new simulated dataset where the $x$ coordinates are given by xobs with uncertainty xerr:
In [20]:
np.random.seed(69302)
xerr = 0.5
xobs = x + xerr * np.random.randn(len(x))
plt.errorbar(xobs, y, xerr=xerr, fmt=".k")
plt.xlabel("x")
plt.ylabel("y");
In [21]:
#
# This is my solution to the challenge problem using emcee
#
def emcee_logprior2(params):
lp = emcee_logprior(params[:3])
if not np.isfinite(lp):
return lp
lp += -0.5 * np.sum((xobs - params[3:])**2) / xerr**2
return lp
def emcee_loglike2(params):
m, b, logs = emcee_get_params(params[:3])
xtrue = params[3:]
model = m * xtrue + b
return -0.5 * np.sum((y - model)**2 / np.exp(2*logs) + 2*logs)
def emcee_logprob2(params):
lp = emcee_logprior2(params)
if not np.isfinite(lp):
return -np.inf
return lp + emcee_loglike2(params)
np.random.seed(123)
init = np.append([0.0, 0.0, 0.0], xobs)
init = init + 0.1 * np.random.randn(100, len(init))
emcee_sampler2 = emcee.EnsembleSampler(init.shape[0], init.shape[1], emcee_logprob2)
state = emcee_sampler2.run_mcmc(init, 30000, progress=True)
thin = 50
emcee_samples2 = emcee_sampler2.get_chain(flat=True, thin=thin, discard=1000)
emcee_samples2 = np.array(emcee_get_params(emcee_samples2[:, :3].T)).T
print("Effective number of samples for each parameter:")
print(thin * len(emcee_samples2) / emcee_sampler2.get_autocorr_time(thin=thin, discard=1000))
corner.corner(emcee_samples2, labels=["m", "b", "logs"],
truths=[m_true, b_true, logs_true]);
In [22]:
#
# This is my solution to the challenge problem using dynesty
#
def dynesty_prior_transform2(u):
bperp = -10 + 20 * u[0]
theta = -0.5*np.pi + np.pi * u[1]
logs = -10 + 20 * u[2]
xtrue = xobs + scipy.stats.norm.ppf(u[3:], scale=xerr)
return np.append([bperp, theta, logs], xtrue)
def dynesty_loglike2(params):
return emcee_loglike2(params)
np.random.seed(543002)
dynesty_sampler2 = dynesty.NestedSampler(dynesty_loglike2, dynesty_prior_transform2, 3 + len(xobs))
dynesty_sampler2.run_nested()
dynesty_results2 = dynesty_sampler2.results
dynesty_samples2 = dynesty.utils.resample_equal(
dynesty_results2.samples, np.exp(dynesty_results2.logwt - dynesty_results2.logz[-1])
)
dynesty_samples2 = np.array(dynesty_get_params(dynesty_samples2[:, :3].T)).T
print("Effective number of samples:")
print(np.any(np.abs(np.diff(np.sort(dynesty_samples2, axis=0), axis=0)) > 1e-10, axis=1).sum())
corner.corner(dynesty_samples2, labels=["m", "b", "logs"],
truths=[m_true, b_true, logs_true]);
In [23]:
#
# This is my solution to the challenge problem using pymc3
#
with pm.Model() as pymc3_model2:
xtrue = pm.Normal("xtrue", mu=xobs, sd=xerr, shape=len(xobs))
bperp = pm.Uniform("bperp", lower=-10, upper=10)
theta = pm.Uniform("theta", lower=-0.5*np.pi, upper=0.5*np.pi)
logs = pm.Uniform("logs", lower=-10, upper=10)
m = pm.Deterministic("m", pm.math.tan(theta))
b = pm.Deterministic("b", bperp / pm.math.cos(theta))
model = m * xtrue + b
pm.Normal("loglike", mu=model, sd=pm.math.exp(logs), observed=y)
np.random.seed(13958)
trace2 = pm.sample(tune=5000, draws=5000)
pymc3_samples2 = pm.trace_to_dataframe(trace2, varnames=["m", "b", "logs"])
corner.corner(pymc3_samples2, truths=[m_true, b_true, logs_true]);
pm.summary(trace2)
Out[23]:
In this example, we'll fit a somewhat more realistic example where we will fit for the periods, phases, and amplitudes of two sine waves in a one dimensional dataset. This isn't terribly compelling, so let's call the independent coordinate "time" and the dependent coordinate "radial velocity", then the problem becomes the estimation of the periods and semi-amplitudes of two planets orbiting a star observed with radial velocities.
First let's simulate the dataset:
In [24]:
# A simple radial velocity model
def rv_model(amps, periods, phases, t, math=np):
t = np.reshape(t, list(np.shape(t)) + [1] * len(np.shape(amps)))
rv = amps * math.sin(2*np.pi*t / periods + phases)
return math.sum(rv, axis=-1)
# Set the true parameters
np.random.seed(21048)
true_periods = np.array([10.5, 15.6])
true_amps = np.array([3.1049, 4.560])
true_phases = np.random.uniform(-np.pi, np.pi, len(true_periods))
# Simulate the data
t = np.random.uniform(0, 100, 78)
v = rv_model(true_amps, true_periods, true_phases, t)
verr = np.random.uniform(1.0, 2.0, len(t))
v = v + verr * np.random.randn(len(t))
plt.errorbar(t, v, yerr=verr, fmt=".k", label="data")
# Plot the true model
t0 = np.linspace(-10, 110, 1000)
v0 = rv_model(true_amps, true_periods, true_phases, t0)
plt.plot(t0, v0, label="true model", lw=1)
plt.xlim(t0.min(), t0.max())
plt.legend(fontsize=10)
plt.xlabel("time")
plt.ylabel("radial velocity");
The likelihood function in this case is:
$$ p(\{v_n\}\,|\,\{t_n\},\,\{A_j\},\,\{P_j\},\,\{\phi_j\}) = \prod_{n=1}^N \mathrm{Normal}(v_n\,|\,\mu_n,\,\sigma_n) $$where
$$ \mu_n = \sum_{j=1}^J A_j\,\sin\left(\frac{2\,\pi\,t_n}{P_j} + \phi_j\right) $$and the parameters are the amplitudes $A_j$, periods $P_j$, and phases $\phi_j$ for each planet.
You should also select reasonable priors for each of these parameters (perhaps uniform or uniform in the log).
Then in the following cells, implement this fit using at least one of the above samplers (or another if you have a favorite). If you have time, it is worth implementing the fit using dynesty and one of the other two samplers to compare your results.
In all cases, some care should be taken to implement periodic boundary conditions for $\phi$. One option is to reparameterize using $u_j = \sqrt{A_j}\,\sin\phi_j$ and $w_j = \sqrt{A_j}\,\cos\phi_j$ instead of $A_j$ and $\phi_j$ directly. (Can you convince yourself that this will have the right prior on $A_j$ and $\phi_j$?) Another option is to parameterize in terms of $A_j$, $u_j \sim \mathrm{Normal}(0,\,1)$, and $w_j \sim \mathrm{Normal}(0,\,1)$ and then define $\phi_j = \arctan(u_j / w_j)$. This is the parameterization that I would use with PyMC3 and it is implemented as the Angle distribution in the advanced_sampling_utils module. If you're using dynesty, you can use the periodic parameter when initializing the NestedSampler.
In [25]:
#
# This is my solution to the RV problem using emcee
#
# Set up the model as above
def emcee_logprior_rv(params):
periods, amps, phases = emcee_get_params_rv(params)
if np.any(periods < 5.0) or np.any(20.0 < periods):
return -np.inf
if np.any(amps < 0.1) or np.any(10.0 < amps):
return -np.inf
return 0.0
def emcee_get_params_rv(params):
log_periods = params[:2]
sqrt_amp_sin_phases = params[2:4]
sqrt_amp_cos_phases = params[4:6]
periods = np.exp(log_periods)
amps = sqrt_amp_sin_phases**2 + sqrt_amp_cos_phases**2
phases = np.arctan2(sqrt_amp_sin_phases, sqrt_amp_cos_phases)
return periods, amps, phases
def emcee_loglike_rv(params):
periods, amps, phases = emcee_get_params_rv(params)
model = rv_model(amps, periods, phases, t)
return -0.5 * np.sum(((v - model) / verr)**2)
def emcee_logprob_rv(params):
lp = emcee_logprior_rv(params)
if not np.isfinite(lp):
return -np.inf
return lp + emcee_loglike_rv(params)
# Initialize somewhere reasonable - how about near the true values? :)
np.random.seed(9281)
init = np.concatenate((
np.log(true_periods),
np.sqrt(true_amps) * np.sin(true_phases),
np.sqrt(true_amps) * np.cos(true_phases))
)
init = init + 1e-6 * np.random.randn(100, len(init))
# Run the sampler
emcee_sampler_rv = emcee.EnsembleSampler(len(init), init.shape[1], emcee_logprob_rv)
emcee_sampler_rv.run_mcmc(init, 15000, progress=True)
# Estimate the number of effective samples
thin = 50
emcee_samples_rv = emcee_sampler_rv.get_chain(flat=True, thin=thin, discard=500)
emcee_samples_rv = np.concatenate(emcee_get_params_rv(emcee_samples_rv.T)).T
print("Effective number of samples for each parameter:")
print(thin * len(emcee_samples_rv) / emcee_sampler_rv.get_autocorr_time(thin=thin, discard=500))
# Plot the corner plot
corner.corner(emcee_samples_rv, labels=["period 1", "period 2", "amp 1", "amp 2", "phase 1", "phase 2"],
truths=np.concatenate((true_periods, true_amps, true_phases)));
# Plot the posterior prediction
plt.figure()
plt.errorbar(t, v, yerr=verr, fmt=".k", label="data")
vsamp = rv_model(emcee_samples_rv[:, 2:4], emcee_samples_rv[:, :2], emcee_samples_rv[:, 4:], t0)
q = np.percentile(vsamp, [16, 50, 84], axis=1)
plt.fill_between(t0, q[0], q[2], color="C1", edgecolor="none", alpha=0.3)
plt.plot(t0, q[1], color="C1", label="fit")
plt.plot(t0, v0, ":k", label="truth", lw=1)
plt.xlim(t0.min(), t0.max())
plt.legend(fontsize=10)
plt.xlabel("time")
plt.ylabel("radial velocity");
In [26]:
#
# This is my solution to the RV problem using dynesty
#
def dynesty_prior_transform_rv(u):
params = np.empty_like(u)
params[:2] = np.log(5) + (np.log(20) - np.log(5)) * u[:2]
params[2:4] = 0.1 + (10 - 0.1) * u[2:4]
params[4:] = 2 * np.pi * (u[4:] % 1.0)
return params
def dynesty_get_params_rv(params):
log_periods = params[:2]
amps = params[2:4]
phases = params[4:]
periods = np.exp(log_periods)
return periods, amps, phases
def dynesty_loglike_rv(params):
periods, amps, phases = dynesty_get_params_rv(params)
model = rv_model(amps, periods, phases, t)
return -0.5 * np.sum(((v - model) / verr)**2)
# Run the sampler
np.random.seed(857930)
dynesty_sampler_rv = dynesty.NestedSampler(dynesty_loglike_rv, dynesty_prior_transform_rv, 6, periodic=[4, 5])
dynesty_sampler_rv.run_nested()
# Resample the chains
dynesty_results_rv = dynesty_sampler_rv.results
dynesty_samples_rv = dynesty.utils.resample_equal(
dynesty_results_rv.samples, np.exp(dynesty_results_rv.logwt - dynesty_results_rv.logz[-1])
)
dynesty_samples_rv = np.concatenate(dynesty_get_params_rv(dynesty_samples_rv.T)).T
# Estimate the number of effective samples
print("Effective number of samples:")
print(np.any(np.abs(np.diff(np.sort(dynesty_samples_rv, axis=0), axis=0)) > 1e-10, axis=1).sum())
# Plot the corner plot
corner.corner(dynesty_samples_rv, labels=["period 1", "period 2", "amp 1", "amp 2", "phase 1", "phase 2"],
truths=np.concatenate((true_periods, true_amps, true_phases % (2*np.pi))));
# Plot the posterior prediction
plt.figure()
plt.errorbar(t, v, yerr=verr, fmt=".k", label="data")
vsamp = rv_model(dynesty_samples_rv[:, 2:4], dynesty_samples_rv[:, :2], dynesty_samples_rv[:, 4:], t0)
q = np.percentile(vsamp, [16, 50, 84], axis=1)
plt.fill_between(t0, q[0], q[2], color="C1", edgecolor="none", alpha=0.3)
plt.plot(t0, q[1], color="C1", label="fit")
plt.plot(t0, v0, ":k", label="truth", lw=1)
plt.xlim(t0.min(), t0.max())
plt.legend(fontsize=10)
plt.xlabel("time")
plt.ylabel("radial velocity");
In [27]:
#
# This is my solution to the RV problem using pymc3
#
from advanced_sampling_utils import Angle
with pm.Model() as pmyc3_model_rv:
# Here I'm constraining the periods to be sorted to break the labelling degeneracy
log_periods = pm.Uniform("log_periods", lower=np.log(5), upper=np.log(20), shape=2,
testval=np.log(true_periods),
transform=pm.distributions.transforms.ordered)
periods = pm.Deterministic("periods", tt.exp(log_periods))
amps = pm.Uniform("amps", lower=0.1, upper=10, shape=2, testval=true_amps)
phases = Angle("phases", shape=2, testval=true_phases)
model = amps[None, :] * tt.sin(2*np.pi*t[:, None] / periods[None, :] + phases[None, :])
model = tt.sum(model, axis=1)
pm.Normal("obs", mu=model, sd=verr, observed=v)
trace_rv = pm.sample(tune=5000, draws=5000, start=pmyc3_model_rv.test_point, target_accept=0.9)
# Plot the corner plot
pymc3_samples_rv = pm.trace_to_dataframe(trace_rv, varnames=["periods", "amps", "phases"])
corner.corner(pymc3_samples_rv, labels=["period 1", "period 2", "amp 1", "amp 2", "phase 1", "phase 2"],
truths=np.concatenate((true_periods, true_amps, true_phases)))
# Plot the posterior prediction
plt.figure()
plt.errorbar(t, v, yerr=verr, fmt=".k", label="data")
vsamp = rv_model(trace_rv["amps"], trace_rv["periods"], trace_rv["phases"], t0)
q = np.percentile(vsamp, [16, 50, 84], axis=1)
plt.fill_between(t0, q[0], q[2], color="C1", edgecolor="none", alpha=0.3)
plt.plot(t0, q[1], color="C1", label="fit")
plt.plot(t0, v0, ":k", label="truth", lw=1)
plt.xlim(t0.min(), t0.max())
plt.legend(fontsize=10)
plt.xlabel("time")
plt.ylabel("radial velocity")
# Check the convergence
pm.summary(trace_rv)
Out[27]:
Try to debug your sampling until you get something convincing. Once you do, try comparing the plots you made to the solutions. When doing that, consider these questions:
In [ ]: