Inverse Transform & Rejection Sampling

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MarcoSC08

Results:

1. $f_1(x)$ over validation function $t_1(x)$ with scipy optimize.fmin.
   

(see nbviewer in Repo Visualization 2-3)

1. Maximization arg max. in LaTeX
   

2. Inverse Transform Method in LaTeX.
   

3. Inverse Transform Method with matplotlib.
   

4. Acceptance-Rejection Method within $f_1(x)$ over $t_1(x)$.
   

Rejected values $\approx$ 0.256

5. Acceptance-Rejection Method of $f_1(x)$ over $t_2(x)$.
   

Rejected values $\approx$ 0.486

6. Acceptance-Rejection Method of $f_2(x)$ over $t_1(x)$ and $f_2(x)$ over $t_3(x)$.
   

Rejected values $\approx$ 0.446

Rejected values $\approx$ 0.506

Considerations:

• Rejection sampling can be used with given/unknown functions $f(x)$ with its densities modelled by KDEs or other methods.
  
• Proposing $t(x)$ can be hard.
  

References:

• Inverse Transform samplings
• Acceptance Rejection samplings
• Pseudo-random number samplings
  
• f(x) Probability density function
  
• F(x) Cumulative distribution function
  
• Uniform distribution
  
• scipy.optimize.fmin
• np.random.random np.random.uniform