Intuition Behind the "Bayesian LASSO"

Intuition Behind the "Bayesian LASSO".carbide.md

Intuition Behind the "Bayesian LASSO"

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HiddenAssumption.js

function HiddenAssumption(Y, SIG_EP_HI, SIG_EP_LO, SIG_BET_HI, SIG_BET_LO, LAM){
    var Values = [];
    for(var n = -1000; n <= 1000; n++){
        var bet = (1/1) * n/1000;
        
    	var PriorBetHi = (1/(Math.sqrt(2 * Math.PI * SIG_BET_HI**2))) * Math.exp(-(bet - 0)**2/(2 * SIG_BET_HI**2));
        var PriorBetLo = (1/(Math.sqrt(2 * Math.PI * SIG_BET_LO**2))) * Math.exp(-(bet - 0)**2/(2 * SIG_BET_LO**2));

        var ProbBetHiEpHi = (1/2) * (LAM/SIG_EP_HI) * Math.exp(- (1/2) * (LAM**2/SIG_EP_HI**2) * SIG_BET_HI**2);
        var ProbBetLoEpHi = (1/2) * (LAM/SIG_EP_HI) * Math.exp(- (1/2) * (LAM**2/SIG_EP_HI**2) * SIG_BET_LO**2);
        
        var ProbBetHiEpLo = (1/2) * (LAM/SIG_EP_LO) * Math.exp(- (1/2) * (LAM**2/SIG_EP_LO**2) * SIG_BET_HI**2);
        var ProbBetLoEpLo = (1/2) * (LAM/SIG_EP_LO) * Math.exp(- (1/2) * (LAM**2/SIG_EP_LO**2) * SIG_BET_LO**2);
        
        var PriorEpHi = (ProbBetHiEpHi/(ProbBetHiEpHi + ProbBetLoEpHi)) * PriorBetHi + (ProbBetLoEpHi/(ProbBetHiEpHi + ProbBetLoEpHi)) * PriorBetLo;
        var PriorEpLo = (ProbBetHiEpLo/(ProbBetHiEpLo + ProbBetLoEpLo)) * PriorBetHi + (ProbBetLoEpLo/(ProbBetHiEpLo + ProbBetLoEpLo)) * PriorBetLo;
        
        var FitEpHi = (1/(Math.sqrt(2 * Math.PI * SIG_EP_HI**2))) * Math.exp(-(Y - bet)**2/(2 * SIG_EP_HI**2));        
        var FitEpLo = (1/(Math.sqrt(2 * Math.PI * SIG_EP_LO**2))) * Math.exp(-(Y - bet)**2/(2 * SIG_EP_LO**2));        
        
        var LikelihoodEpHi = FitEpHi * PriorEpHi;
        var LikelihoodEpLo = FitEpLo * PriorEpLo;
        
        var LogLikelihood = Math.log(LikelihoodEpHi * (50/100) + LikelihoodEpLo * (50/100));
        var MaxValue      = 10;
        if (LogLikelihood < MaxValue) {
            Values.push(LogLikelihood);
        } else {
            Values.push(MaxValue);
        }
	}
    return Values
}


var y       = 107/1000;
var sigEpH  = 100;
var sigEpL  = 0.50;
var sigBetH = 100;
var sigBetL = 0.50;
var lam     = 2.5313;

var Data = HiddenAssumption(y, sigEpH, sigEpL, sigBetH, sigBetL, lam);

MixtureModel.js

function MixtureModel(Y, SIG_EP, SIG_BET_H, SIG_BET_L, TET){
    var Values = [];
    for(var n = -100; n <= 100; n++){
        var bet = n/100;
        
    	var PriorH = 1/(Math.sqrt(2 * Math.PI * SIG_BET_H**2)) * Math.exp(-(bet - 0)**2/(2 * SIG_BET_H**2));
        var PriorL = 1/(Math.sqrt(2 * Math.PI * SIG_BET_L**2)) * Math.exp(-(bet - 0)**2/(2 * SIG_BET_L**2));
        var Prior  = TET * PriorH + (1-TET) * PriorL;
        
        var Fit = 1/(Math.sqrt(2 * Math.PI * SIG_EP**2)) * Math.exp(-(Y - bet)**2/(2 * SIG_EP**2));        
        
        var LogLikelihood = Math.log(Fit * Prior);
        if (LogLikelihood < 10) {
            Values.push(LogLikelihood);
        } else {
            Values.push(10);
        }
	}
    return Values
}

var y       = 0.3438;
var sigEp   = 0.3594;
var sigBetH = 3.4688;
var sigBetL = 0.1406;
var tet     = 0.0625;
var LogLikelihood = MixtureModel(y, sigEp, sigBetH, sigBetL, tet);

StandardModel.js

function StandardModel(Y, SIG_EP, SIG_BET){
    var Values = [];
    for(var n = -100; n <= 100; n++){
        var bet = n/100;
        
    	var Fit           = 1/(Math.sqrt(2 * Math.PI * SIG_EP**2)) * Math.exp(-(Y - bet)**2/(2 * SIG_EP**2));
        var Prior         = 1/(Math.sqrt(2 * Math.PI * SIG_BET**2)) * Math.exp(-(bet - 0)**2/(2 * SIG_BET**2));
        var LogLikelihood = Math.log(Fit * Prior);
        if (LogLikelihood < 1) {
            Values.push(LogLikelihood);
        } else {
        	Values.push(1);    
        }
	}
    return Values
}

var y      = 0.3197;
var sigEp  = 0.4268;
var sigBet = 0.5083;
var LogLikelihood = StandardModel(y, sigEp, sigBet);
var betHat = (sigBet**2/(sigEp**2 + sigBet**2)) * y;