Add more about evidence and selection to flow (#142)

* Add Laplace evidence

* Numerically stable laplace evidence

* Minor edits to Laplace

* Remove rmax

* Rm old things

* Rm comments

* Add script

* Add super toy selection

* Add super toy selection

* Update script

scripts/flow/test_harmonic.py

@@ -0,0 +1,60 @@
+from argparse import ArgumentParser, ArgumentTypeError
+
+
+def parse_args():
+    parser = ArgumentParser()
+    parser.add_argument("--device", type=str, default="cpu",
+                        help="Device to use.")
+    return parser.parse_args()
+
+
+ARGS = parse_args()
+# This must be done before we import JAX etc.
+from numpyro import set_host_device_count, set_platform                         # noqa
+
+set_platform(ARGS.device)                                                       # noqa
+
+from jax import numpy as jnp                                                    # noqa
+import numpy as np                                                              # noqa
+import csiborgtools                                                             # noqa
+from scipy.stats import multivariate_normal                                     # noqa
+
+
+def get_harmonic_evidence(samples, log_posterior, nchains_harmonic, epoch_num):
+    """Compute evidence using the `harmonic` package."""
+    data, names = csiborgtools.dict_samples_to_array(samples)
+    data = data.reshape(nchains_harmonic, -1, len(names))
+    log_posterior = log_posterior.reshape(nchains_harmonic, -1)
+
+    return csiborgtools.harmonic_evidence(
+        data, log_posterior, return_flow_samples=False, epochs_num=epoch_num)
+
+
+ndim = 250
+nsamples = 100_000
+nchains_split = 10
+loc = jnp.zeros(ndim)
+cov = jnp.eye(ndim)
+
+
+gen = np.random.default_rng()
+X = gen.multivariate_normal(loc, cov, size=nsamples)
+samples = {f"x_{i}": X[:, i] for i in range(ndim)}
+logprob = multivariate_normal(loc, cov).logpdf(X)
+
+neg_lnZ_laplace, neg_lnZ_laplace_error = csiborgtools.laplace_evidence(
+    samples, logprob, nchains_split)
+print(f"neg_lnZ_laplace:  {neg_lnZ_laplace} +/- {neg_lnZ_laplace_error}")
+
+
+neg_lnZ_harmonic, neg_lnZ_harmonic_error = get_harmonic_evidence(
+    samples, logprob, nchains_split, epoch_num=30)
+print(f"neg_lnZ_harmonic: {neg_lnZ_harmonic} +/- {neg_lnZ_harmonic_error}")
+
+
+
+
+
+
+
+