def plot_wikipedia_pdfs():
"""
https://en.wikipedia.org/wiki/Normal_distribution#/media/File:Normal_Distribution_PDF.svg
"""
ax = new_axes()
Normal(mu=0, sigma_sq=0.2).plot(x=x, color='blue', ax=ax)
Normal(mu=0, sigma_sq=1.0).plot(x=x, color='red', ax=ax)
Normal(mu=0, sigma_sq=5.0).plot(x=x, color='orange', ax=ax)
Normal(mu=-2, sigma_sq=0.5).plot(x=x, color='green', ax=ax)
ax.legend(loc='upper right')
plt.show()
def likelihood(self, **kwargs) -> Normal:
"""
Return a distribution reflecting the likelihood of observing
the data, under a Normal model, independent of the prior belief
about the distribution of parameter μ.
"""
return Normal(mu=self._x_sum / self._n, sigma_sq=self._sigma_sq)
def prior_predictive(self) -> Normal:
"""
Return a Normal describing the expected distribution of future
values based on the prior parameter estimates.
"""
return Normal(mu=self.mu_0, sigma_sq=self.sigma_0_sq +
self.sigma_sq).with_y_label(r'$P(\tilde{X}=x|'
r'μ_{Norm}=μ_{0, Norm},'
r'σ²=σ²_{0, Norm}+σ²)$')
def prior(self) -> Normal:
"""
Return a Normal distribution reflecting the prior belief about the
distribution of the parameter μ, before seeing any data.
"""
if self._parameterization == 'σ²':
y_label = ('$P(μ_{Norm}=x|' 'μ_{0, Norm},' 'σ²_{0, Norm})$')
else:
y_label = ('$P(μ_{Norm}=x|' 'μ_{0, Norm},' 'τ_{0, Norm})$')
return Normal(mu=self.mu_0, sigma_sq=self.sigma_0_sq).with_x_label(
'μ').with_y_label(y_label).prepend_to_label('Prior: ')
def posterior(self) -> Normal:
"""
Return a Normal distribution reflecting the posterior belief about the
distribution of the parameter μ, after observing the data.
"""
if self._parameterization == 'σ²':
y_label = ('$P(μ_{Norm}=x|' 'μ_{0, Norm}′,' 'σ²_{0, Norm}′)$')
else:
y_label = ('$P(μ_{Norm}=x|' 'μ_{0, Norm}′,' 'τ_{0, Norm}′)$')
return Normal(
mu=self.mu_0_prime, sigma_sq=self.sigma_0_sq_prime).with_x_label(
'μ').with_y_label(y_label).prepend_to_label('Posterior: ')
def posterior(self) -> Normal:
return Normal(mu=self.mu_0_prime,
sigma_sq=self.sigma_sq_0_prime).with_x_label(
'μ').prepend_to_label('Posterior: ')
def prior(self) -> Normal:
return Normal(mu=self.mu_0, sigma_sq=self.sigma_sq_0).with_x_label(
'μ').prepend_to_label('Prior: ')