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function [a, b] = beta_specification ( mu, sigma2, lb, ub, name)
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% Returns the hyperparameters of the beta distribution given the expectation and variance.
%
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% INPUTS
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% - mu [double] Expectation of the Gamma random variable.
% - sigma2 [double] Variance of the Gamma random variable.
% - lb [double] Lower bound of the domain (default is zero).
% - ub [double] Upper bound of the domain (default is one).
%
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% OUTPUTS
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% - a [double] First hyperparameter of the Beta density.
% - b [double] Second hyperparameter of the Beta density.
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% Copyright © 2015-2023 Dynare Team
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%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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if nargin < 3
lb = 0 ;
ub = 1 ;
end
if nargin < 4
ub = 1 ;
end
if nargin > 4 && ~ isempty ( name )
name1 = sprintf ( ' of %s ' , name ) ;
name2 = sprintf ( ' (for %s)' , name ) ;
else
name1 = ' ' ;
name2 = ' ' ;
end
if mu < lb
error ( [ ' The prior expectation (%f) %scannot be smaller than the lower bound of the Beta distribution (%f)!' ] , mu , name1 , lb )
end
if mu > ub
error ( ' The prior expectation (%f) %scannot be greater than the upper bound of the Beta distribution (%f)!' , mu , name1 , ub )
end
len = ub - lb ;
mu = ( mu - lb ) / len ;
sigma2 = sigma2 / ( len * len ) ;
if sigma2 > ( 1 - mu ) * mu
error ( ' Beta prior%s. Given the declared prior expectation, prior lower and upper bounds, the prior std. has to be smaller than %f.' , name2 , sqrt ( ( 1 - mu ) * mu ) )
end
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a = ( 1 - mu ) * mu * mu / sigma2 - mu ;
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b = a * ( 1 / mu - 1 ) ;
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return % --*-- Unit tests --*--
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%@test:1
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try
[ a , b ] = beta_specification ( . 5 , . 05 ) ;
t ( 1 ) = true ;
catch
t ( 1 ) = false ;
end
if t ( 1 )
t ( 2 ) = abs ( 0.5 - a / ( a + b ) ) < 1e-12 ;
t ( 3 ) = abs ( 0.05 - a * b / ( ( a + b ) ^2 * ( a + b + 1 ) ) ) < 1e-12 ;
end
T = all ( t ) ;
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%@eof:1
%@test:2
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try
[ a , b ] = beta_specification ( 0.5 , . 05 , 1 , 2 ) ;
t ( 1 ) = false ;
catch
t ( 1 ) = true ;
end
T = all ( t ) ;
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%@eof:2
%@test:3
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try
[ a , b ] = beta_specification ( 2.5 , . 05 , 1 , 2 ) ;
t ( 1 ) = false ;
catch
t ( 1 ) = true ;
end
T = all ( t ) ;
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%@eof:3
%@test:4
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try
[ a , b ] = beta_specification ( . 4 , . 6 * . 4 + 1e-4 ) ;
t ( 1 ) = false ;
catch
t ( 1 ) = true ;
end
T = all ( t ) ;
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%@eof:4
%@test:5
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try
[ a , b ] = beta_specification ( . 4 , . 6 * . 4 + 1e-6 ) ;
t ( 1 ) = false ;
catch
t ( 1 ) = true ;
end
try
[ a , b ] = beta_specification ( . 4 , . 6 * . 4 - 1e-6 ) ;
t ( 2 ) = true ;
catch
t ( 2 ) = false ;
end
T = all ( t ) ;
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%@eof:5