IMP::isd::PenalizedComplexityPrior Class Reference

Penalized complexity prior. More...

#include <IMP/isd/PenalizedComplexityPrior.h>

Inheritance diagram for IMP::isd::PenalizedComplexityPrior:

Detailed Description

Penalized complexity prior.

Given a parameter \(\xi\) of a model (distribution) \(f(x \mid \xi)\) with a simpler base model \(g(x)\) ( \(f(x \mid \xi=0)\) if \(\xi\) is a flexibility parameter), the Penalized Complexity (PC) prior on \(\xi\) minimizes the complexity introduced by deviation from the base model. It corresponds to an exponential distribution on the Kullback–Leibler divergence between the flexible and base models, in terms of \(\xi\).

For ease of parameterization, a 'tail event' is defined as an upper bound on an interpretable transformation \(Q(\xi)\) with tail density \(\alpha\), such that \(P(Q(\xi) > U) = \alpha\). With KL divergence defined as

\[ \mathrm{KLD}(f(x \mid \xi) || g(x)) = \int f(x \mid \xi) \log\left(\frac{f(x \mid \xi)}{g(x)}\right) dx,\]

KL divergence at \(\xi\) normalized by KL divergence at the tail

\[ D(\xi) = \frac{\mathrm{KLD}(f(x \mid \xi) || g(x))}{\mathrm{KLD}(f(x \mid \xi=Q^{-1}(U)) || g(x))} ,\]

and

\[ J(\xi) = \frac{\partial D(\xi)}{\partial \xi} ,\]

the density of the PC prior is

\[p(\xi) = \frac{-\log\alpha}{2 \sqrt{D(\xi)}} \alpha^{\sqrt{D(\xi)}} |J(\xi)|.\]

See Also

NormalSigmaPCRestraint

Note

For more details, see Simpson et al. Stat Sci. (2017) 32(1): 1. arXiv: 1403.4630

The log-density with this parameterization overflows when \(D(\xi)=0\). The value of the PC prior at the base model is unique to each case and cannot be provided by this general implementation.

Definition at line 52 of file PenalizedComplexityPrior.h.

Public Member Functions
	PenalizedComplexityPrior (double Dxi, double Jxi, double alpha, std::string name="PenalizedComplexityPrior %1%")
	Create from normalized KL divergence. More...
double	evaluate_derivative_alpha () const
	Evaluate derivative of negative log-density wrt alpha. More...
double	evaluate_derivative_Dxi () const
	Evaluate derivative of negative log-density wrt \(D(\xi)\). More...
double	evaluate_derivative_Jxi () const
	Evaluate derivative of negative log-density wrt derivative of \(D(\xi)\). More...
double	evaluate_derivative_xi (double JJxi) const
	Evaluate derivative of negative log-density wrt parameter \(\xi\). More...
double	get_alpha () const
	Get density in tail. More...
double	get_Dxi () const
	Get normalized KL divergence \(D(\xi)\). More...
double	get_Jxi () const
	Get derivative of normalized KL divergence \(D(\xi)\) wrt \(\xi\). More...
virtual std::string	get_type_name () const
virtual ::IMP::VersionInfo	get_version_info () const
	Get information about the module and version of the object. More...
void	set_alpha (double v)
	Set density in tail. More...
void	set_Dxi (double v)
	Set normalized KL divergence \(D(\xi)\). More...
void	set_Jxi (double v)
	Set derivative of normalized KL divergence \(D(\xi)\) wrt \(\xi\). More...
void	update_sufficient_statistics (Floats Dxis, Floats Jxis)
	Update sufficient statistics with values and derivatives. More...
Public Member Functions inherited from IMP::isd::OneDimensionalSufficientDistribution
	OneDimensionalSufficientDistribution (std::string name="OneDimensionalSufficientDistribution %1%")
	Constructor. More...
double	evaluate () const
	Get negative log-density using cached sufficient statistics. More...
double	get_density () const
	Get probability density using cached sufficient statistics. More...
Floats	get_sufficient_statistics () const
void	update_sufficient_statistics (Floats vs)
	Update cached sufficient statistics from data. More...
Public Member Functions inherited from IMP::isd::Distribution
	Distribution (std::string name="Distribution %1%")
Public Member Functions inherited from IMP::Object
virtual void	clear_caches ()
CheckLevel	get_check_level () const
LogLevel	get_log_level () const
void	set_check_level (CheckLevel l)
void	set_log_level (LogLevel l)
	Set the logging level used in this object. More...
void	set_was_used (bool tf) const
void	show (std::ostream &out=std::cout) const
const std::string &	get_name () const
void	set_name (std::string name)

Protected Member Functions
virtual double	do_evaluate () const
virtual Floats	do_get_sufficient_statistics () const
virtual void	do_update_sufficient_statistics (Floats Dxis)
virtual void	do_update_sufficient_statistics (Floats Dxis, Floats Jxis)
Protected Member Functions inherited from IMP::isd::OneDimensionalSufficientDistribution
virtual double	do_get_density () const
Protected Member Functions inherited from IMP::Object
	Object (std::string name)
	Construct an object with the given name. More...
virtual void	do_destroy ()

Constructor & Destructor Documentation

IMP::isd::PenalizedComplexityPrior::PenalizedComplexityPrior	(	double	Dxi,
		double	Jxi,
		double	alpha,
		std::string	name = "PenalizedComplexityPrior %1%"
	)

Create from normalized KL divergence.

Parameters

[in]	Dxi	Normalized KL divergence \(D(\xi)\).
[in]	Jxi	Derivative \(J(\xi)\) of normalized KL divergence.
[in]	alpha	Density in tail.
[in]	name	Name of prior.

Member Function Documentation

double IMP::isd::PenalizedComplexityPrior::evaluate_derivative_alpha

(

)

const

Evaluate derivative of negative log-density wrt alpha.

double IMP::isd::PenalizedComplexityPrior::evaluate_derivative_Dxi

(

)

const

Evaluate derivative of negative log-density wrt \(D(\xi)\).

double IMP::isd::PenalizedComplexityPrior::evaluate_derivative_Jxi

(

)

const

Evaluate derivative of negative log-density wrt derivative of \(D(\xi)\).

double IMP::isd::PenalizedComplexityPrior::evaluate_derivative_xi

(

double

JJxi

)

const

Evaluate derivative of negative log-density wrt parameter \(\xi\).

Parameters

[in]

JJxi

Second derivative of normalized KL divergence wrt \(\xi\).

double IMP::isd::PenalizedComplexityPrior::get_alpha

(

)

const

Get density in tail.

double IMP::isd::PenalizedComplexityPrior::get_Dxi

(

)

const

Get normalized KL divergence \(D(\xi)\).

double IMP::isd::PenalizedComplexityPrior::get_Jxi

(

)

const

Get derivative of normalized KL divergence \(D(\xi)\) wrt \(\xi\).

virtual ::IMP::VersionInfo IMP::isd::PenalizedComplexityPrior::get_version_info ( ) const

virtual

Get information about the module and version of the object.

Reimplemented from IMP::isd::OneDimensionalSufficientDistribution.

Definition at line 106 of file PenalizedComplexityPrior.h.

void IMP::isd::PenalizedComplexityPrior::set_alpha

(

double

v

)

Set density in tail.

void IMP::isd::PenalizedComplexityPrior::set_Dxi

(

double

v

)

Set normalized KL divergence \(D(\xi)\).

void IMP::isd::PenalizedComplexityPrior::set_Jxi

(

double

v

)

Set derivative of normalized KL divergence \(D(\xi)\) wrt \(\xi\).

void IMP::isd::PenalizedComplexityPrior::update_sufficient_statistics	(	Floats	Dxis,
		Floats	Jxis
	)

Update sufficient statistics with values and derivatives.

Definition at line 70 of file PenalizedComplexityPrior.h.

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The documentation for this class was generated from the following file:

• PenalizedComplexityPrior.h