Bayesian Models¶

Bernoulli distribution¶

class cprior.BernoulliModel(alpha=1, beta=1)¶

Bases: cprior.cdist.beta.BetaModel

Bayesian model with a Bernoulli likelihood and a beta prior distribution.

Given data samples $\mathbf{x} = (x_1, \ldots, x_n)$ from a Bernoulli distribution with parameter $p$, the posterior distribution is

\[p | \mathbf{x} \sim \mathcal{B}(\alpha + \sum_{i=1}^n x_i, \beta + n - \sum_{i=1}^n x_i).\]

with prior parameters $\alpha$ and $\beta$.

Parameters:	alpha (int or float (default=1)) – Prior parameter alpha. beta (int or float (default=1)) – Prior parameter beta.

n_success_¶: int – Number of successes.

n_samples_¶: int – Number of samples.

alpha_posterior¶

Posterior parameter alpha.

Returns:	alpha
Return type:	float

beta_posterior¶

Posterior parameter beta.

Returns:	beta
Return type:	float

cdf(x)¶

Cumulative distribution function of the posterior distribution.

Parameters:	x (array-like) – Quantiles.
Returns:	cdf – Cumulative distribution function evaluated at x.
Return type:	numpy.ndarray

credible_interval()¶

Credible interval of the posterior distribution.

Parameters:	interval_length (float (default=0.9)) – Compute `interval_length`% credible interval. This is a value in [0, 1].
Returns:	interval – Lower and upper credible interval limits.
Return type:	tuple

mean()¶: Mean of the posterior distribution.

pdf(x)¶

Probability density function of the posterior distribution.

Parameters:	x (array-like) – Quantiles.
Returns:	pdf – Probability density function evaluated at x.
Return type:	numpy.ndarray

ppf(q)¶

Percent point function (quantile) of the posterior distribution.

Parameters:	x (array-like) – Lower tail probability.
Returns:	ppf – Quantile corresponding to the lower tail probability q.
Return type:	numpy.ndarray

ppmean()¶

Posterior predictive mean.

If $X$ is a Bernoulli trial with parameter $p \sim Beta(\alpha, \beta)$, then the posterior predictive expected value is given by

\[\mathrm{E}[X] = \frac{\alpha}{\alpha + \beta},\]

where $\alpha$ and $\beta$ are the posterior values of the parameters.

Returns:	mean
Return type:	float

pppdf(x)¶

Posterior predictive probability density function.

If $X$ is a Bernoulli trial with parameter $p \sim Beta(\alpha, \beta)$, then the posterior predictive probability density function is given by

\[\begin{split}f(x; \alpha, \beta) = \begin{cases} \frac{\alpha}{\alpha+ \beta} & \text{if $x = 1$}\\ \frac{\beta}{\alpha+ \beta} & \text{if $x = 0$} \end{cases}\end{split}\]

where $\alpha$ and $\beta$ are the posterior values of the parameters.

Returns:	pdf – Probability density function evaluated at x.
Return type:	float

ppvar()¶

Posterior predictive variance.

If $X$ is a Bernoulli trial with parameter $p \sim Beta(\alpha, \beta)$, then the posterior predictive variance is given by

\[\mathrm{Var}[X] = \frac{\alpha \beta}{(\alpha + \beta)^2},\]

where $\alpha$ and $\beta$ are the posterior values of the parameters.

Returns:	var
Return type:	float

rvs(size=1, random_state=None)¶

Random variates of the posterior distribution.

Parameters:	size (int (default=1)) – Number of random variates. random_state (int or None (default=None)) – The seed used by the random number generator.
Returns:	rvs – Random variates of given size.
Return type:	numpy.ndarray or scalar

std()¶: Standard deviation of the posterior distribution.

update(data)¶

Update posterior parameters with new data samples.

Parameters:	data (array-like, shape = (n_samples)) – Data samples from a Bernoulli distribution.

var()¶: Variance of the posterior distribution.

class cprior.BernoulliABTest(modelA, modelB, simulations=1000000, random_state=None)¶

Bases: cprior.cdist.beta.BetaABTest

Bernoulli A/B test.

Parameters:	modelA (object) – The control model. modelB (object) – The variation model. simulations (int or None (default=1000000)) – Number of Monte Carlo simulations. random_state (int or None (default=None)) – The seed used by the random number generator.

expected_loss(method='exact', variant='A', lift=0)¶

Compute the expected loss. This is the expected uplift lost by choosing a given variant.

If variant == "A", $\mathrm{E}[\max(B - A - lift, 0)]$
If variant == "B", $\mathrm{E}[\max(A - B - lift, 0)]$
If variant == "all", both.

If lift is positive value, the computation method must be Monte Carlo sampling.

Parameters:	method (str (default="exact")) – The method of computation. Options are “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. lift (float (default=0.0)) – The amount of uplift.

expected_loss_ci(method='MC', variant='A', interval_length=0.9)¶

Compute credible intervals on the difference distribution of $Z = B-A$ and/or $Z = A-B$.

If variant == "A", $Z = B - A$
If variant == "B", $Z = A - B$
If variant == "all", both.

Parameters:	method (str (default="MC")) – The method of computation. Options are “asymptotic” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. interval_length (float (default=0.9)) – Compute `interval_length`% credible interval. This is a value in [0, 1].

expected_loss_relative(method='exact', variant='A')¶

Compute expected relative loss for choosing a variant. This can be seen as the negative expected relative improvement or uplift.

If variant == "A", $\mathrm{E}[(B - A) / A]$
If variant == "B", $\mathrm{E}[(A - B) / B]$
If variant == "all", both.

Parameters:	method (str (default="exact")) – The method of computation. Options are “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”.

expected_loss_relative_ci(method='MC', variant='A', interval_length=0.9)¶

Compute credible intervals on the relative difference distribution of $Z = (B-A)/A$ and/or $Z = (A-B)/B$.

If variant == "A", $Z = (B-A)/A$
If variant == "B", $Z = (A-B)/B$
If variant == "all", both.

Parameters:	method (str (default="MC")) – The method of computation. Options are “asymptotic”, “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. interval_length (float (default=0.9)) – Compute `interval_length`% credible interval. This is a value in [0, 1].

probability(method='exact', variant='A', lift=0)¶

Compute the error probability or chance to beat control.

If variant == "A", $P[A > B + lift]$
If variant == "B", $P[B > A + lift]$
If variant == "all", both.

If lift is positive value, the computation method must be Monte Carlo sampling.

Parameters:	method (str (default="exact")) – The method of computation. Options are “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. lift (float (default=0.0)) – The amount of uplift.

Binomial distribution¶

Geometric distribution¶

class cprior.GeometricModel(alpha=1, beta=1)¶

Bases: cprior.cdist.beta.BetaModel

Bayesian model with geometric likelihood and a beta prior distribution.

Parameters:	alpha (int or float (default=1)) – Prior parameter alpha. beta (int or float (default=1)) – Prior parameter beta.

alpha_posterior¶

Posterior parameter alpha.

Returns:	alpha
Return type:	float

beta_posterior¶

Posterior parameter beta.

Returns:	beta
Return type:	float

cdf(x)¶

Cumulative distribution function of the posterior distribution.

Parameters:	x (array-like) – Quantiles.
Returns:	cdf – Cumulative distribution function evaluated at x.
Return type:	numpy.ndarray

credible_interval()¶

Credible interval of the posterior distribution.

Parameters:	interval_length (float (default=0.9)) – Compute `interval_length`% credible interval. This is a value in [0, 1].
Returns:	interval – Lower and upper credible interval limits.
Return type:	tuple

mean()¶: Mean of the posterior distribution.

pdf(x)¶

Probability density function of the posterior distribution.

Parameters:	x (array-like) – Quantiles.
Returns:	pdf – Probability density function evaluated at x.
Return type:	numpy.ndarray

ppf(q)¶

Percent point function (quantile) of the posterior distribution.

Parameters:	x (array-like) – Lower tail probability.
Returns:	ppf – Quantile corresponding to the lower tail probability q.
Return type:	numpy.ndarray

rvs(size=1, random_state=None)¶

Random variates of the posterior distribution.

Parameters:	size (int (default=1)) – Number of random variates. random_state (int or None (default=None)) – The seed used by the random number generator.
Returns:	rvs – Random variates of given size.
Return type:	numpy.ndarray or scalar

std()¶: Standard deviation of the posterior distribution.

update(data)¶

Update posterior parameters with new data samples.

Parameters:	data (array-like, shape = (n_samples)) – Data samples from a geometric distribution.

var()¶: Variance of the posterior distribution.

class cprior.GeometricABTest(modelA, modelB, simulations=1000000, random_state=None)¶

Bases: cprior.cdist.beta.BetaABTest

Geometric A/B test.

Parameters:	modelA (object) – The control model. modelB (object) – The variation model. simulations (int or None (default=1000000)) – Number of Monte Carlo simulations. random_state (int or None (default=None)) – The seed used by the random number generator.

expected_loss(method='exact', variant='A', lift=0)¶

Compute the expected loss. This is the expected uplift lost by choosing a given variant.

If variant == "A", $\mathrm{E}[\max(B - A - lift, 0)]$
If variant == "B", $\mathrm{E}[\max(A - B - lift, 0)]$
If variant == "all", both.

If lift is positive value, the computation method must be Monte Carlo sampling.

Parameters:	method (str (default="exact")) – The method of computation. Options are “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. lift (float (default=0.0)) – The amount of uplift.

expected_loss_ci(method='MC', variant='A', interval_length=0.9)¶

Compute credible intervals on the difference distribution of $Z = B-A$ and/or $Z = A-B$.

If variant == "A", $Z = B - A$
If variant == "B", $Z = A - B$
If variant == "all", both.

Parameters:	method (str (default="MC")) – The method of computation. Options are “asymptotic” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. interval_length (float (default=0.9)) – Compute `interval_length`% credible interval. This is a value in [0, 1].

expected_loss_relative(method='exact', variant='A')¶

Compute expected relative loss for choosing a variant. This can be seen as the negative expected relative improvement or uplift.

If variant == "A", $\mathrm{E}[(B - A) / A]$
If variant == "B", $\mathrm{E}[(A - B) / B]$
If variant == "all", both.

Parameters:	method (str (default="exact")) – The method of computation. Options are “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”.

expected_loss_relative_ci(method='MC', variant='A', interval_length=0.9)¶

Compute credible intervals on the relative difference distribution of $Z = (B-A)/A$ and/or $Z = (A-B)/B$.

If variant == "A", $Z = (B-A)/A$
If variant == "B", $Z = (A-B)/B$
If variant == "all", both.

Parameters:	method (str (default="MC")) – The method of computation. Options are “asymptotic”, “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. interval_length (float (default=0.9)) – Compute `interval_length`% credible interval. This is a value in [0, 1].

probability(method='exact', variant='A', lift=0)¶

Compute the error probability or chance to beat control.

If variant == "A", $P[A > B + lift]$
If variant == "B", $P[B > A + lift]$
If variant == "all", both.

If lift is positive value, the computation method must be Monte Carlo sampling.

Parameters:	method (str (default="exact")) – The method of computation. Options are “exact” and “MC”. variant (str (default="A")) – The chosen variant. Options are “A”, “B”, “all”. lift (float (default=0.0)) – The amount of uplift.

Bayesian Models¶

Bernoulli distribution¶

Binomial distribution¶

Geometric distribution¶

Negative binomial distribution¶