# 5. Post Analysis¶

The output of an inference scheme is a Journal (abcpy.output.Journal) which holds all the necessary results and convenient methods to do the post analysis.

## Basis Analysis¶

One can easily access the sampled parameters and corresponding weights using:

print(journal.get_parameters())
print(journal.get_weights())


The output of get_parameters() is a Python dictionary. The keys for this dictionary are the names you specified for the parameters. The corresponding values are the marginal posterior samples of that parameter. Here is a short example of what you would specify, and what would be the output in the end:

a = Normal([[1],[0.1]], name='parameter_1')
b = MultivariateNormal([[1,1],[[0.1,0],[0,0.1]]], name='parameter_2')


If one defined a model with these two parameters as inputs and n_sample=2, the following would be the output of journal.get_parameters():

{'parameter_1' : [[0.95],[0.97]], 'parameter_2': [[0.98,1.03],[1.06,0.92]]}


These are samples at the final step of ABC algorithm. If you want samples from the earlier steps of a sequential algorithm you can get a Python dictionary for that step by using:

journal.get_parameters(step_number)


Since this is a dictionary, you can also access the values for each step as:

journal.get_parameters(step_number)["name"]


For the post analysis basic functions are provided:

# do post analysis
print(journal.posterior_mean())
print(journal.posterior_cov())


Also, to ensure reproducibility, every journal stores the parameters of the algorithm that created it:

print(journal.configuration)


And certainly, a journal can easily be saved to and loaded from disk:

journal.save("experiments.jnl")
new_journal = Journal.fromFile('experiments.jnl')


## Posterior plots and diagnostics¶

You can plot the inferred posterior distribution of the parameters in the following way:

journal.plot_posterior_distr(path_to_save="posterior.png")


The above line plots the posterior distribution for all the parameters and stores it in posterior.png; if you instead want to plot it for some parameters only, you can use the parameters_to_show argument; in addition, the ranges_parameters argument can be used to provide a dictionary specifying the limits for the axis in the plots:

journal.plot_posterior_distr(parameters_to_show='parameter_1',
ranges_parameters={'parameter_1': [0,2]})


For journals generated with sequential algorithms, we provide a way to check the convergence by plotting the estimated Effective Sample Size (ESS) at each iteration, as well as an estimate of the Wasserstein distance between the empirical distributions defined by the samples and weights at subsequent iterations:

journal.plot_ESS()
journal.Wass_convergence_plot()


Instead, for journals generated by MCMC, we provide way to plot the traceplot for the required parameters:

journal.traceplot()


## Posterior resampling and predictive check¶

In some cases, you may want to resample (for instance, bootstrapping or subsampling) the posterior samples stored in a Journal, by tacking into account the posterior weights. This can be done using the resample() method. Behind the scenes, this uses the numpy.random.choice method, and it inherits arguments from it. It allows to do different things, for instance:

• if the set of posterior samples (weighted or unweighted) is too large, you can obtained a subsampled (without replacement) set by doing:
new_journal = journal.resample(n_samples=100, replace=False)

• Alternatively, if the used algorithm returns weighted posterior samples, you may want instead an unweighted set of samples obtained by sampling with replacement (commonly called bootstrapping); this can be done with the following line (where the number of required bootstrapped samples in the new journal is unspecified and therefore corresponding to the number of samples in the old journal):
new_journal = journal.resample()


Finally, in some cases you may want to generate simulations from the model for parameter values sampled from the posterior, for instance in order to check similarity with the original observation (predictive check). ABCpy provides the output.GenerateFromJournal to do that. This class needs to be instanstiated by providing to it the model and the backend which you want to use for the simulation; then, you can pass a Journal as argument to the generate() method in order to generate simulations from the posterior samples contained there:

generate_from_journal = GenerateFromJournal([model], backend=backend)
parameters, simulations, normalized_weights = generate_from_journal.generate(journal)