IMP Reference Guide  develop.0ea7f7a4db,2020/10/23 The Integrative Modeling Platform
IMP::isd::FStudentT Class Reference

Joint Student's t-distribution of Function. More...

#include <IMP/isd/FStudentT.h>

Inheritance diagram for IMP::isd::FStudentT:

## Detailed Description

Joint Student's t-distribution of Function.

If a vector of values $$F(X)$$ is jointly distributed according to the t distribution with shared center $$F(M)$$, scale $$\sigma$$, and degrees of freedom $$\nu$$, then the vector $$X$$ is distributed according to the F-t distribution, where $$F$$ is a one-to-one (strictly monotonic) function.

For a vector $$X=\{x_1,...,x_N\}$$ containing $$N$$ observations, the joint F-t density is given by

$p(X \mid M, \sigma, \nu) = \frac{ \Gamma[(N + \nu) / 2]}{\Gamma[\nu / 2]} (\pi \nu)^{-N / 2} \sigma^{-N} J(X) \left[1 + \frac{t(X, M, \sigma)^2}{\nu} \right]^{-(N + \nu)/2},$

where

$J(X) = \left| \prod_{i=1}^N F'(x_i) \right|,$

is a normalization factor according to the change of variables technique.

$t(X, M, \sigma)^2 = \frac{S_2(X) - 2 F(M) S_1(X) + N F(M)^2}{\sigma^2},$

and the minimally sufficient statistics are

\begin{align*} S_1(X) &= \sum_{i=1}^N F(x_i) \ S_2(X) &= \sum_{i=1}^N F(x_i)^2. \end{align*}

The degrees of freedom $$\nu$$ controls the heaviness of the tails. When $$\nu = 1$$, the F-t distribution becomes the F-Cauchy distribution. For very large $$\nu$$, the F-t distribution approaches the F-Normal distribution. Set $$J(X)$$ values to 1 for the joint t distribution.

Note
The joint t distribution is derived by marginalizing a joint normal distribution with the scaled inverse chi-square distribution as a prior on the variance. The same distribution is not produced by multiplying t distributions. For more details, see Gelmen et al. Bayesian Data Analaysis. 3rd edition.
FNormal

Definition at line 61 of file FStudentT.h.

## Public Member Functions

FStudentT (Floats FXs, Floats JXs, double FM, double sigma, double nu, std::string name="FStudentT %1%")
Create from observations vectors. More...

FStudentT (double sumFX, double sumFX2, unsigned N, double LogJX, double FM, double sigma, double nu, std::string name="FStudentT %1%")
Create from sufficient statistics. More...

virtual double evaluate_derivative_FM () const
Evaluate derivative of negative log-density wrt $$F(M)$$. More...

virtual double evaluate_derivative_Fx (double Fx) const
Evaluate derivative of negative log-density wrt single $$F(x_i)$$. More...

Floats evaluate_derivative_FX (const Floats FXs) const
Evaluate derivative of negative log-density wrt elements of $$F(X)$$. More...

virtual double evaluate_derivative_LogJX () const
Evaluate derivative of negative log-density wrt $$\log J(X)$$. More...

virtual double evaluate_derivative_nu () const
Evaluate derivative of negative log-density wrt $$\nu$$. More...

virtual double evaluate_derivative_sigma () const
Evaluate derivative of negative log-density wrt $$\sigma$$. More...

double get_FM ()

double get_LogJX ()

unsigned get_N ()

double get_nu ()

double get_sigma ()

double get_sumFX ()
Get sufficient statistic for passed $$J(X)$$ values. More...

double get_sumFX2 ()

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_FM (double v)

void set_LogJX (double v)

void set_N (unsigned v)

void set_nu (double v)

void set_sigma (double v)

void set_sumFX (double v)

void set_sumFX2 (double v)

void update_cached_values () const
Update cached intermediate values. More...

void update_sufficient_statistics (Floats FXs, Floats JXs)
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 FXs)

virtual void do_update_sufficient_statistics (Floats FXs, Floats JXs)

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::FStudentT::FStudentT ( Floats FXs, Floats JXs, double FM, double sigma, double nu, std::string name = "FStudentT %1%" )

Create from observations vectors.

Parameters
 [in] FXs Vector of N $$F(x)$$ observations drawn independently from the same F-t distribution. [in] JXs Vector of N derivatives of $$F(x)$$ with respect to $$X$$ vector. [in] FM Center of F-Student t with respect to $$F(x)$$. [in] sigma Scale of F-Student t with respect to $$F(x)$$. [in] nu Degrees of freedom of Student t distribution. [in] name Name.
 IMP::isd::FStudentT::FStudentT ( double sumFX, double sumFX2, unsigned N, double LogJX, double FM, double sigma, double nu, std::string name = "FStudentT %1%" )

Create from sufficient statistics.

Parameters
 [in] sumFX Sum of observations of $$F(x)$$. [in] sumFX2 Sum of observations of $$F(x)^2$$. [in] N Number of observations. [in] LogJX Log of $$J(X)$$. [in] FM Center of F-t with respect to $$F(x)$$. [in] sigma Scale of F-t with respect to $$F(x)$$. [in] nu Degrees of freedom of Student t distribution. [in] name Name.

## Member Function Documentation

 virtual double IMP::isd::FStudentT::evaluate_derivative_FM ( ) const
virtual

Evaluate derivative of negative log-density wrt $$F(M)$$.

 virtual double IMP::isd::FStudentT::evaluate_derivative_Fx ( double Fx ) const
virtual

Evaluate derivative of negative log-density wrt single $$F(x_i)$$.

$$F(x_i)$$ represents an element of $$F(X)$$. Since only the sufficient statistics are stored, this value must be provided.

 Floats IMP::isd::FStudentT::evaluate_derivative_FX ( const Floats FXs ) const

Evaluate derivative of negative log-density wrt elements of $$F(X)$$.

This is equivalent to though faster than running evaluate_derivative_Fx() on every element of $$F(X)$$.

 virtual double IMP::isd::FStudentT::evaluate_derivative_LogJX ( ) const
virtual

Evaluate derivative of negative log-density wrt $$\log J(X)$$.

 virtual double IMP::isd::FStudentT::evaluate_derivative_nu ( ) const
virtual

Evaluate derivative of negative log-density wrt $$\nu$$.

 virtual double IMP::isd::FStudentT::evaluate_derivative_sigma ( ) const
virtual

Evaluate derivative of negative log-density wrt $$\sigma$$.

 double IMP::isd::FStudentT::get_sumFX ( )

Get sufficient statistic for passed $$J(X)$$ values.

Definition at line 141 of file FStudentT.h.

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

Get information about the module and version of the object.

Reimplemented from IMP::isd::OneDimensionalSufficientDistribution.

Definition at line 149 of file FStudentT.h.

 void IMP::isd::FStudentT::update_cached_values ( ) const

Update cached intermediate values.

This method is automatically called during evaluation. If evaluating derivatives after modifying parameters but before evaluation, this method must be manually called.

 void IMP::isd::FStudentT::update_sufficient_statistics ( Floats FXs, Floats JXs )

Update sufficient statistics with values and derivatives.

Definition at line 97 of file FStudentT.h.

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