# Normal distribution¶

The normal distribution or Gaussian distribution is a continuous probability distribution. The probability density function of a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$ for $$x \in \mathbb{R}$$ is

$f(x; \mu, \sigma) = \frac{\exp\left(-\frac{1}{2} \left(\frac{x-\mu}{\sigma}\right)^2\right)}{\sigma\sqrt{2 \pi}},$

and the cumulative distribution is

$F(x; \mu, \sigma) = \frac{1}{2}\left(1 + \mathrm{erf}\left(\frac{x-\mu} {\sigma\sqrt{2}}\right)\right).$

The expected value and variance are as follows

$\mathrm{E}[X] = \mu. \quad \mathrm{Var}[X] = \sigma^2.$

The normal distribution is used to model/approximate symmetric centralized distributions.

class cprior.models.NormalModel(name='', loc=0.001, variance_scale=0.001, shape=0.001, scale=0.001)

Bases: cprior.cdist.normal_inverse_gamma.NormalInverseGammaModel

Bayesian model with a normal likelihood and a normal-inverse-gamma prior distribution.

Given data samples $$\mathbf{x} = (x_1, \ldots, x_n)$$ from a normal distribution with parameters mean $$\mu$$, and variance $$\sigma^2$$, the posterior distribution is

$\mu, \sigma^2 | \mathbf{x} \sim \mathcal{N}\Gamma^{-1}\left(\mu_n, \lambda_n, \alpha_n, \beta_n\right),$

where,

\begin{align}\begin{aligned}\mu_n &= \frac{\lambda \mu_0 + n \bar{x}}{\lambda + n},\\\lambda_n &= \lambda + n,\\\alpha_n &= \alpha + \frac{n}{2},\\\beta_n &= \beta + \frac{1}{2} \left(\sum_{i=1}^n (x_i - \bar{x})^2 + \frac{n \lambda (\bar{x} - \mu_0)^2}{\lambda + n} \right).\end{aligned}\end{align}

with prior parameters $$\mu_0$$ (loc), $$\lambda$$ (variance_scale), $$\alpha$$ (shape) and $$\beta$$ (scale). Note that $$n \bar{x} = \sum_{i=1}^n x_i$$.

Parameters: name (str (default="")) – Model name. loc (float (default=0.001)) – Prior parameter loc. variance_scale (float (default=0.001)) – Prior parameter variance_scale. shape (float (default=0.001)) – Prior parameter shape. scale (float (default=0.001)) – Prior parameter scale.
n_samples_

Number of samples.

Type: int
cdf(x, sig2)

Cumulative distribution function of the posterior distribution.

Parameters: x (array-like) – Quantiles. sig2 (array-like) – Quantiles. cdf – Cumulative distribution function evaluated at (x, sig2). numpy.ndarray
credible_interval(interval_length)

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]. interval – Lower and upper credible interval limits. tuple
loc_posterior

Posterior parameter mu (location).

Returns: mu float
mean()

Mean of the posterior distribution.

Returns: mean tuple of floats
pdf(x, sig2)

Probability density function of the posterior distribution.

Parameters: x (array-like) – Quantiles. sig2 (array-like) – Quantiles. pdf – Probability density function evaluated at (x, sig2). numpy.ndarray
ppf(q)

Percent point function (quantile) of the posterior distribution.

Parameters: x (array-like) – Lower tail probability. ppf – Quantile corresponding to the lower tail probability q. tuple of numpy.ndarray
ppmean()

Posterior predictive mean.

If $$X$$ follows a normal distribution with parameters $$\mu$$ and $$\sigma^2$$, then the posterior predictive expected value is given by

$\mathrm{E}[X] = \mu_0,$

where $$\mu_0$$ is the posterior value of the parameter.

Returns: mean float
pppdf(x)

Posterior predictive probability density function.

If $$X$$ follows a normal distribution with parameters $$\mu$$ and $$\sigma^2$$, then the posterior predictive probability density function is given by the probability density function of the following Student’s t-distribution

$t_{2 \alpha}\left(\mu_0, \frac{\beta (1 + \lambda^{-1})}{\alpha}\right),$

where $$\mu_0$$, $$\lambda$$, $$\alpha$$ and $$\beta$$ are the posterior values of the parameters.

Parameters: x (array-like) – Quantiles. pdf – Probability density function evaluated at x. float
ppvar()

Posterior predictive variance.

If $$X$$ follows a normal distribution with parameters $$\mu$$ and $$\sigma^2$$, then the posterior predictive variance is given by

$\mathrm{Var}[X] = \frac{\left(\beta(1 + \lambda^{-1})\right)^2} {\alpha(\alpha - 1)},$

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

Returns: var 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. rvs – Random variates of given size (size, 2). numpy.ndarray
scale_posterior

Posterior parameter beta (scale).

Returns: beta float
shape_posterior

Posterior parameter alpha (shape).

Returns: alpha float
std()

Standard deviation of the posterior distribution.

Returns: std tuple of floats
update(data)

Update posterior parameters with new data.

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

Variance of the posterior distribution.

Returns: var tuple of floats
variance_scale_posterior

Posterior parameter lambda (variance_scale).

Returns: lambda float
class cprior.models.NormalABTest(modelA, modelB, simulations=1000000, random_state=None)

Bases: cprior.cdist.normal_inverse_gamma.NormalInverseGammaABTest

Normal 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 tuple of floats

Notes

Method “exact” uses the normal approximation of the Student’s t-distribution for the expected loss of the mean when the number of degrees of freedom is large. For small values, numerical intergration is used.

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

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]. ci_method (str (default="ETI")) – Method to compute credible intervals. Supported methods are Highest Density interval (method="HDI) and Equal-tailed interval (method="ETI"). Currently, method="HDI is only available for method="MC". expected_loss_ci tuple of floats
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 tuple of floats

Notes

Method “exact” uses an approximation of $$E[1/X]$$ where $$X$$ follows a Student’s t-distribution.

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

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]. ci_method (str (default="ETI")) – Method to compute credible intervals. Supported methods are Highest Density interval (method="HDI) and Equal-tailed interval (method="ETI"). Currently, method="HDI is only available for method="MC". expected_loss_relative_ci tuple of floats

Notes

Method “exact” uses the normal approximation of the Student’s t-distribution for the expected loss of the mean.

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. probability tuple of floats

Notes

Method “exact” uses the normal approximation of the Student’s t-distribution for the error probability of the mean when the number of degrees of freedom is large. For small values, numerical intergration is used.

update_A(data)

Update posterior parameters for variant A with new data samples.

Parameters: data (array-like, shape = (n_samples)) –
update_B(data)

Update posterior parameters for variant B with new data samples.

Parameters: data (array-like, shape = (n_samples)) –
class cprior.models.NormalMVTest(models, simulations=1000000, random_state=None, n_jobs=None)

Bases: cprior.cdist.normal_inverse_gamma.NormalInverseGammaMVTest

Normal Multivariate test.

Parameters: models (dict) – The control and variations models. 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', control='A', variant='B', lift=0)

Compute the expected loss. This is the expected uplift lost by choosing a given variant, i.e., $$\mathrm{E}[\max(control - variant - lift, 0)]$$.

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”. control (str (default="A")) – The control variant. variant (str (default="B")) – The tested variant. lift (float (default=0.0)) – The amount of uplift. expected_loss tuple of floats

Notes

Method “exact” uses the normal approximation of the Student’s t-distribution for the expected loss of the mean when the number of degrees of freedom is large. For small values, numerical intergration is used.

expected_loss_ci(method='MC', control='A', variant='B', interval_length=0.9, ci_method='ETI')

Compute credible intervals on the difference distribution of $$Z = control-variant$$.

Parameters: method (str (default="MC")) – The method of computation. Options are “asymptotic” and “MC”. control (str (default="A")) – The control variant. variant (str (default="B")) – The tested variant. interval_length (float (default=0.9)) – Compute interval_length% credible interval. This is a value in [0, 1]. ci_method (str (default="ETI")) – Method to compute credible intervals. Supported methods are Highest Density interval (method="HDI) and Equal-tailed interval (method="ETI"). Currently, method="HDI is only available for method="MC". expected_loss_ci tuple of floats
expected_loss_relative(method='exact', control='A', variant='B')

Compute expected relative loss for choosing a variant. This can be seen as the negative expected relative improvement or uplift, i.e., $$\mathrm{E}[(control - variant) / variant]$$.

Parameters: method (str (default="exact")) – The method of computation. Options are “exact” and “MC”. control (str (default="A")) – The control variant. variant (str (default="B")) – The tested variant. expected_loss_relative tuple of floats

Notes

Method “exact” uses an approximation of $$E[1/X]$$ where $$X$$ follows a Student’s t-distribution.

expected_loss_relative_ci(method='MC', control='A', variant='B', interval_length=0.9, ci_method='ETI')

Compute credible intervals on the relative difference distribution of $$Z = (control - variant) / variant$$.

Parameters: method (str (default="MC")) – The method of computation. Options are “asymptotic”, “exact” and “MC”. control (str (default="A")) – The control variant. variant (str (default="B")) – The tested variant. interval_length (float (default=0.9)) – Compute interval_length% credible interval. This is a value in [0, 1]. ci_method (str (default="ETI")) – Method to compute credible intervals. Supported methods are Highest Density interval (method="HDI) and Equal-tailed interval (method="ETI"). Currently, method="HDI is only available for method="MC". expected_loss_relative_ci tuple of floats
expected_loss_relative_vs_all(method='quad', control='A', variant='B', mlhs_samples=1000)

Compute the expected relative loss against all variations. For example, given variants “A”, “B”, “C” and “D”, and choosing variant=”B”, we compute $$\mathrm{E}[(\max(A, C, D) - B) / B]$$.

Parameters: method (str (default="quad")) – The method of computation. Options are “MC” (Monte Carlo), “MLHS” (Monte Carlo + Median Latin Hypercube Sampling) and “quad” (numerical integration). variant (str (default="B")) – The chosen variant. mlhs_samples (int (default=1000)) – Number of samples for MLHS method. expected_loss_relative_vs_all tuple of floats
expected_loss_vs_all(method='quad', variant='B', lift=0, mlhs_samples=1000)

Compute the expected loss against all variations. For example, given variants “A”, “B”, “C” and “D”, and choosing variant=”B”, we compute $$\mathrm{E}[\max(\max(A, C, D) - B, 0)]$$.

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

Parameters: method (str (default="quad")) – The method of computation. Options are “MC” (Monte Carlo), “MLHS” (Monte Carlo + Median Latin Hypercube Sampling) and “quad” (numerical integration). variant (str (default="B")) – The chosen variant. lift (float (default=0.0)) – The amount of uplift. mlhs_samples (int (default=1000)) – Number of samples for MLHS method. expected_loss_vs_all tuple of floats
probability(method='exact', control='A', variant='B', lift=0)

Compute the error probability or chance to beat control, i.e., $$P[variant > control + lift]$$.

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”. control (str (default="A")) – The control variant. variant (str (default="B")) – The tested variant. lift (float (default=0.0)) – The amount of uplift. probability tuple of floats

Notes

Method “exact” uses the normal approximation of the Student’s t-distribution for the error probability of the mean when the number of degrees of freedom is large. For small values, numerical intergration is used.

probability_vs_all(method='quad', variant='B', lift=0, mlhs_samples=1000)

Compute the error probability or chance to beat all variations. For example, given variants “A”, “B”, “C” and “D”, and choosing variant=”B”, we compute $$P[B > \max(A, C, D) + lift]$$.

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

Parameters: method (str (default="quad")) – The method of computation. Options are “MC” (Monte Carlo), “MLHS” (Monte Carlo + Median Latin Hypercube Sampling) and “quad” (numerical integration). variant (str (default="B")) – The chosen variant. lift (float (default=0.0)) – The amount of uplift. mlhs_samples (int (default=1000)) – Number of samples for MLHS method. probability_vs_all tuple of floats
update(data, variant)

Update posterior parameters for a given variant with new data samples.

Parameters: data (array-like, shape = (n_samples)) – variant (str) –