Review core examples

• Add a README
• Move non-dynesty examples into separate sub-directory to avoid repeatition
• Standardize variable names
• Remove the multidimensional gaussian: the 15d example is better and more complete
• Remove the start_mcmc_chains_near.. example: this is demonstrating a rarely used feature of a rarely used sampler, better to put this into the docs if needed.

examples/core_examples/alternative_samplers/linear_regression_bilby_mcmc.py

+#!/usr/bin/env python
+"""
+An example of how to use bilby to perform parameter estimation for
+non-gravitational wave data. In this case, fitting a linear function to
+data with background Gaussian noise
+
+"""
+import bilby
+import matplotlib.pyplot as plt
+import numpy as np
+
+# A few simple setup steps
+label = "linear_regression_bilby_mcmc"
+outdir = "outdir"
+bilby.utils.check_directory_exists_and_if_not_mkdir(outdir)
+
+
+# First, we define our "signal model", in this case a simple linear function
+def model(time, m, c):
+    return time * m + c
+
+
+# Now we define the injection parameters which we make simulated data with
+injection_parameters = dict(m=0.5, c=0.2)
+
+# For this example, we'll use standard Gaussian noise
+
+# These lines of code generate the fake data. Note the ** just unpacks the
+# contents of the injection_parameters when calling the model function.
+sampling_frequency = 10
+time_duration = 10
+time = np.arange(0, time_duration, 1 / sampling_frequency)
+N = len(time)
+sigma = np.random.normal(1, 0.01, N)
+data = model(time, **injection_parameters) + np.random.normal(0, sigma, N)
+
+# We quickly plot the data to check it looks sensible
+fig, ax = plt.subplots()
+ax.plot(time, data, "o", label="data")
+ax.plot(time, model(time, **injection_parameters), "--r", label="signal")
+ax.set_xlabel("time")
+ax.set_ylabel("y")
+ax.legend()
+fig.savefig("{}/{}_data.png".format(outdir, label))
+
+# Now lets instantiate a version of our GaussianLikelihood, giving it
+# the time, data and signal model
+likelihood = bilby.likelihood.GaussianLikelihood(time, data, model, sigma)
+
+# From hereon, the syntax is exactly equivalent to other bilby examples
+# We make a prior
+priors = dict()
+priors["m"] = bilby.core.prior.Uniform(0, 5, "m")
+priors["c"] = bilby.core.prior.Uniform(-2, 2, "c")
+
+# And run sampler
+result = bilby.run_sampler(
+    likelihood=likelihood,
+    priors=priors,
+    sampler="bilby_mcmc",
+    nsamples=10000,
+    ntemps=1,
+    nensemble=1,
+    L1steps=1,
+    print_dt=5,
+    injection_parameters=injection_parameters,
+    outdir=outdir,
+    label=label,
+)
+
+# Finally plot a corner plot: all outputs are stored in outdir
+result.plot_corner()