---

jupytext:

text_representation:

extension: .myst

extension: .md

format_name: myst

format_version: 0.13

jupytext_version: 1.13.8

jupytext_version: 1.16.4

kernelspec:

display_name: Python 3 (ipykernel)

language: python

As usual, we begin by importing some Python code.

```{code-cell} ipython3

:tags: [hide-output]

from numpyro.infer import ELBO as nELBO

from numpyro.infer import Trace_ELBO as nTrace_ELBO

from numpyro.optim import Adam as nAdam

```

## Unleashing MCMC on a Binomial Likelihood

This lecture begins with the binomial example in the {doc}`quantecon lecture <prob_meaning>`.

- The truncated Laplace can be created using `Numpyro`'s `TruncatedDistribution` class.

```{code-cell} ipython3

# used by Numpyro

def TruncatedLogNormal_trans(loc, scale):

Computes numerically the posterior distribution with beta prior parametrized by (alpha0, beta0)

given data using MCMC

"""

# tensorize

data = torch.tensor(data)

# use pyro

if self.solver=='pyro':

# tensorize

data = torch.tensor(data)

nuts_kernel = NUTS(self.model)

mcmc = MCMC(nuts_kernel, num_samples=num_samples, warmup_steps=num_warmup, disable_progbar=True)

mcmc.run(data)

# use numpyro

elif self.solver=='numpyro':

data = np.array(data, dtype=float)

nuts_kernel = nNUTS(self.model)

mcmc = nMCMC(nuts_kernel, num_samples=num_samples, num_warmup=num_warmup, progress_bar=False)

mcmc.run(self.rng_key, data=data)

params : the learned parameters for guide

losses : a vector of loss at each step

"""

# tensorize data

if not torch.is_tensor(data):

data = torch.tensor(data)

# initiate SVI

svi = self.SVI_init(guide_dist=guide_dist)

# do gradient steps

if self.solver=='pyro':

# tensorize data

if not torch.is_tensor(data):

data = torch.tensor(data)

# store loss vector

losses = np.zeros(n_steps)

for step in range(n_steps):

}

elif self.solver=='numpyro':

data = np.array(data, dtype=float)

result = svi.run(self.rng_key, n_steps, data, progress_bar=False)

params = dict(

(key, np.asarray(value)) for key, value in result.params.items()

Let's start with the analytical method that we described in this quantecon lecture <https://python.quantecon.org/prob_meaning.html>

```{code-cell} ipython3

# First examine Beta priors

BETA_pyro = BayesianInference(param=(5,5), name_dist='beta', solver='pyro')

(Increasing the step size increases computational time though).

```{code-cell} ipython3

BayesianInferencePlot(true_theta, num_list, BETA_numpyro).SVI_plot(guide_dist='beta', n_steps=100000)

```

## Non-conjugate Prior Distributions

Having assured ourselves that our MCMC and VI methods can work well when we have conjugate prior and so can also compute analytically, we

example_CLASS = BayesianInference(param=(0,1), name_dist='uniform', solver='numpyro')

print(f'=======INFO=======\nParameters: {example_CLASS.param}\nPrior Dist: {example_CLASS.name_dist}\nSolver: {example_CLASS.solver}')

BayesianInferencePlot(true_theta, num_list, example_CLASS).SVI_plot(guide_dist='normal', n_steps=SVI_num_steps)

```

```{code-cell} ipython3