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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:
 An example of customising Pints plots
This example builds on adaptive covariance MCMC and pairwise scatterplots, and shows you how to plot the parameter distributions with examples of customising the plots.
Setting up an MCMC routine
See the adaptive covariance MCMC example for details.
Step1:  Plotting Pints' standard 1d histograms
We can now run the MCMC routine and plot the histograms of the inferred parameters.
Step2:  Customise the plots
For example, here our toy model is a logistic model of population growth
$$f(t) = \frac{k}{1 + (k/p_0 - 1)\exp(-rt)},$$
where $r$ is the growth rate, $k$ is the carrying capacity, and $p_0$ is the initial population (fixed constant). In this example, we have model parameters $r$ and $k$, together with a noise parameter $\sigma$, to be inferred.
We will do the following update to the plots to make it prettier
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Python Code:
import pints
import pints.toy as toy
import numpy as np
import matplotlib.pyplot as plt
# Load a forward model
model = toy.LogisticModel()
# Create some toy data
real_parameters = [0.015, 500]
times = np.linspace(0, 1000, 100)
org_values = model.simulate(real_parameters, times)
# Add noise
noise = 50
values = org_values + np.random.normal(0, noise, org_values.shape)
real_parameters = np.array(real_parameters + [noise])
# Get properties of the noise sample
noise_sample_mean = np.mean(values - org_values)
noise_sample_std = np.std(values - org_values)
# Create an object with links to the model and time series
problem = pints.SingleOutputProblem(model, times, values)
# Create a log-likelihood function (adds an extra parameter!)
log_likelihood = pints.GaussianLogLikelihood(problem)
# Create a uniform prior over both the parameters and the new noise variable
log_prior = pints.UniformLogPrior(
    [0.01, 400, noise*0.1],
    [0.02, 600, noise*100]
    )
# Create a posterior log-likelihood (log(likelihood * prior))
log_posterior = pints.LogPosterior(log_likelihood, log_prior)
# Perform sampling using MCMC, with a single chain
x0 = real_parameters * 1.1
mcmc = pints.MCMCController(log_posterior, 1, [x0])
mcmc.set_max_iterations(6000)
mcmc.set_log_to_screen(False)
Explanation: An example of customising Pints plots
This example builds on adaptive covariance MCMC and pairwise scatterplots, and shows you how to plot the parameter distributions with examples of customising the plots.
Setting up an MCMC routine
See the adaptive covariance MCMC example for details.
End of explanation
print('Running...')
chains = mcmc.run()
print('Done!')
# Select chain 0 and discard warm-up
chain = chains[0]
chain = chain[3000:]
import pints.plot
# Plot the 1d histogram of each parameter
pints.plot.histogram([chain])
plt.show()
Explanation: Plotting Pints' standard 1d histograms
We can now run the MCMC routine and plot the histograms of the inferred parameters.
End of explanation
# Plot the 1d histogram of each parameter
fig, axes = pints.plot.histogram([chain])
# Customise the plots
parameter_names = [r'$r$', r'$k$', r'$\sigma$']
for i, ax in enumerate(axes):
    # (1) Add parameter name
    ax.set_xlabel(parameter_names[i])
    # (2i) Add mean
    ax.axvline(np.mean(chain[:, i]), color='k', label='Mean')
    # (2ii) Add 95% credible interval
    ax.axvline(np.percentile(chain[:, i], 2.5), color='C1', label='95% credible interval')
    ax.axvline(np.percentile(chain[:, i], 97.5), color='C1')
axes[0].legend()
# (3) Update figure size
fig.set_size_inches(14, 9)
plt.show()
Explanation: Customise the plots
For example, here our toy model is a logistic model of population growth
$$f(t) = \frac{k}{1 + (k/p_0 - 1)\exp(-rt)},$$
where $r$ is the growth rate, $k$ is the carrying capacity, and $p_0$ is the initial population (fixed constant). In this example, we have model parameters $r$ and $k$, together with a noise parameter $\sigma$, to be inferred.
We will do the following update to the plots to make it prettier:
1. We may want to change the labels to the names of parameters;
2. We will add lines for (i) the mean and (ii) the 95% credible interval;
3. We will make the figure size a bit bigger.
End of explanation
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