def prior(self) -> Gamma:
"""
Return a Gamma distribution reflecting the prior belief about the
distribution of the parameter λ, before seeing any data.
"""
return Gamma(alpha=self._alpha, beta=self._beta).with_y_label(
'$P(λ_{Poi}=x|'
'α_{Gam},'
'β_{Gam})$').prepend_to_label('Prior: ')
def posterior(self) -> Gamma:
"""
Return a Gamma distribution reflecting the posterior belief about the
distribution of the parameter λ, after observing the data.
"""
return Gamma(alpha=self.alpha_prime,
beta=self.beta_prime).with_y_label(
r'$P(λ_{Poi}=x|'
r'α_{Gam}+k_{Obs},'
r'β_{Gam}+n_{Obs})$').prepend_to_label('Posterior: ')
def prior(self) -> Gamma:
return Gamma(
alpha=self._alpha,
beta=self._beta).with_x_label('λ').prepend_to_label('Prior: ')
def posterior(self) -> Gamma:
return Gamma(alpha=self.alpha_prime,
beta=self.beta_prime).with_x_label('λ').prepend_to_label(
'Posterior: ')
def plot_wikipedia_cdfs():
"""
https://en.wikipedia.org/wiki/Gamma_distribution#/media/File:Gamma_distribution_cdf.svg
"""
ax = new_axes()
Gamma.from_k_theta(k=1, theta=2).cdf().plot(x=x, color='red', ax=ax)
Gamma.from_k_theta(k=2, theta=2).cdf().plot(x=x, color='orange', ax=ax)
Gamma.from_k_theta(k=3, theta=2).cdf().plot(x=x, color='yellow', ax=ax)
Gamma.from_k_theta(k=5, theta=1).cdf().plot(x=x, color='green', ax=ax)
Gamma.from_k_theta(k=9, theta=0.5).cdf().plot(x=x, color='black', ax=ax)
Gamma.from_k_theta(k=7.5, theta=1).cdf().plot(x=x, color='blue', ax=ax)
Gamma.from_k_theta(k=0.5, theta=1).cdf().plot(x=x, color='purple', ax=ax)
ax.set_ylim(0, 1)
ax.legend(loc='lower right')
plt.show()