‹
Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Figure 1.1: Illustration of the curse of dimensionality for density estimation. (Data compression) Figure 1.2: Rejection ABC (Rejection ABC) Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Figure 1.3: Likelihood-based Metropolis–Hastings (Markov-chain Monte Carlo ABC) Figure 1.4: Pseudo-marginal Metropolis–Hastings (Markov-chain Monte Carlo ABC) Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Figure 1.5: Markov-chain Monte Carlo ABC (Markov-chain Monte Carlo ABC) Figure 1.6: Importance-sampling ABC (Sequential Monte Carlo ABC) Figure 1.7: Sequential Monte Carlo ABC (Sequential Monte Carlo ABC) Figure 1.8: Comparison between SNPE-B, SNL and MaxVar for three different simulator models. The plot is taken from (?). (Comparison with active-learning methods) Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

Density estimation: what and why?

$$$$ p br(x) == Pr br(x __deriv in B_epsilon mathopen(x) mathclose) /abs(B_epsilon mathopen(x) mathclose) for * epsilon = 0

Density estimation: what and why?

$$$$ p br(x) >= 0 (for all) x inR**D, int(p br(x) d x) == Pr br(x __deriv in X) (for all Lebesgue - measurable) X * subseteq * R**D

Density estimation: what and why?

$$$$ int(p br(x) d x) == 1

›