estimating satellite collision probabilities via the adaptive splitting technique … ·...
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❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡
❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❋r♦♠ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❚❤❡♦r②
❘✉❞② P❆❙❚❊▲✱ ❖◆❊❘❆✴❉❈P❙✲P❛❧❛✐s❡❛✉❙✉♣❡r✈✐s♦r ✿ ❏érô♠❡ ▼❖❘■❖✱ ❖◆❊❘❆✴❉❈P❙✲P❛❧❛✐s❡❛✉
❙❝✐❡♥t✐✜❝ ❞✐r❡❝t♦r ✿ ❋r❛♥ç♦✐s ▲❡ ●▲❆◆❉✱ ■◆❘■❆ ❞❡ ❘❡♥♥❡s
❏❛♥✉❛r② ✷✵✶✶
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ♠✐❣❤t t❤✐♥❦ ♦❢ ❛♥ ❡♠♣t② ♦✉t❡r s♣❛❝❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ♠✐❣❤t t❤✐♥❦ ♦❢ ❛♥ ❡♠♣t② ♦✉t❡r s♣❛❝❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ✇♦✉❧❞ ❜❡ ✇r♦♥❣✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥❡ ✇♦✉❧❞ ❜❡ ✇r♦♥❣✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❙♣❛❝❡ ❞❡❜r✐s s✉rr♦✉♥❞ ❊❛rt❤ ❛♥❞ ❛❝t✐✈❡ ♦r❜✐t✐♥❣ ♦❜❥❡❝ts✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥ ❚✉❡s❞❛② ❋❡❜r✉❛r② t❤❡ ✶✵t❤ ✷✵✵✾✱ s❛t❡❧❧✐t❡s ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❝♦❧❧✐❞❡❞✳
❲❤❛t ✇❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ✐t ❤❛♣♣❡♥❡❞❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■♥tr♦❞✉❝t✐♦♥
❖♥ ❚✉❡s❞❛② ❋❡❜r✉❛r② t❤❡ ✶✵t❤ ✷✵✵✾✱ s❛t❡❧❧✐t❡s ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❝♦❧❧✐❞❡❞✳
❲❤❛t ✇❛s t❤❡ ♣r♦❜❛❜✐❧✐t② ✐t ❤❛♣♣❡♥❡❞❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s s✐③✐♥❣✿ ❘♦✉❣❤ ❡st✐♠❛t❡s
❘❡❛❧ s✐③❡s ❛r❡ ❝♦♥✜❞❡♥t✐❛❧✳
❋♦r ❝♦♥✈❡♥✐❡♥❝❡ s❛❦❡✱ ✇❡ ❛ss✉♠❡ s♣❤❡r✐❝❛❧ s❛t❡❧❧✐t❡s✳
❘❛❞✐✐ ❞❡❝✐❞❡❞ ❜❛s❡❞ ♦♥ ✜❡❧❞ ❡①♣❡r✐❡♥❝❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ❧♦❝❛❧✐s❛t✐♦♥✿ ❚✇♦ ❧✐♥❡ ❡❧❡♠❡♥t ✭❚▲❊✮
Pr♦✈✐❞❡❞ ❜② t❤❡ ◆❖❘❆❉ ❛♥❞ t❤❡ ◆❆❙❆✳
P♦s✐t✐♦♥ ❛♥❞ ♠♦t✐♦♥ ❛t ♠❡❛s✉r❡♠❡♥t t✐♠❡ ✇✐t❤♦✉t ❛❝❝✉r❛❝②❞❡t❛✐❧s✳
❍♦♠❡ ♠❛❞❡ ♠❡❛♥s ♦❢ t❤❡ ♦r❜✐t ♣❛r❛♠❡t❡rs✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
❙❛t❡❧❧✐t❡s ♣r♦♣❛❣❛t✐♦♥✿ ❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s
❋❛st t♦ ❝♦♠♣✉t❡✳
❊❛s② t♦ ✐♠♣❧❡♠❡♥t✳
▲❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ◆❆❙❆ ♠♦❞❡❧✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
❙✐③✐♥❣✱ ❧♦❝❛❧✐s❛t✐♦♥ ❛♥❞ ♣r♦♣❛❣❛t✐♦♥
■r✐❞✐✉♠✲❈♦s♠♦s ❞✐st❛♥❝❡ ❛❜♦✉t ❝♦❧❧✐s✐♦♥ t✐♠❡ ❛❝❝♦r❞✐♥❣ t♦ ❚▲❊
16.5 16.55 16.6 16.65 16.7 16.75500
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5500
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Iridium−Cosmos Distance on February 10th
2009
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
❘❛♥❞♦♠♥❡ss ❛t ❤❛♥❞✿ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s ✐s ❞❡t❡r♠✐♥✐st✐❝ ♦♥❝❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❛r❡ ❦♥♦✇♥✳
❲❡ ♥♦✐s❡❞ t❤❡ ❚▲❊ t♦ ❝♦♣❡ ✇✐t❤ t❤❡✐r ✉♥❦♥♦✇♥ ✉♥❝❡rt❛✐♥t②✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
❘❛♥❞♦♠♥❡ss ❛t ❤❛♥❞✿ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s ✐s ❞❡t❡r♠✐♥✐st✐❝ ♦♥❝❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❛r❡ ❦♥♦✇♥✳
❲❡ ♥♦✐s❡❞ t❤❡ ❚▲❊ t♦ ❝♦♣❡ ✇✐t❤ t❤❡✐r ✉♥❦♥♦✇♥ ✉♥❝❡rt❛✐♥t②✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
❘❛♥❞♦♠♥❡ss ❛t ❤❛♥❞✿ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❑❡♣❧❡r✐❛♥ ❞②♥❛♠✐❝s ✐s ❞❡t❡r♠✐♥✐st✐❝ ♦♥❝❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s
❛r❡ ❦♥♦✇♥✳
❲❡ ♥♦✐s❡❞ t❤❡ ❚▲❊ t♦ ❝♦♣❡ ✇✐t❤ t❤❡✐r ✉♥❦♥♦✇♥ ✉♥❝❡rt❛✐♥t②✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s t♠ ✐ r♠ ✐ ✈♠ ✐ ✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ ♠ ✐ ♠ ✐ ✵✻ t ✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ ♠ ✐ ♠ ✐ ✵✻ t ✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❞❝♦❧ ❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
∆ (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) = ♠✐♥t∈■
{‖~❘✶(t) − ~❘✷(t)‖}
❞❝♦❧ ❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② t♦ ❡st✐♠❛t❡
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
∆ (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) = ♠✐♥t∈■
{‖~❘✶(t) − ~❘✷(t)‖}
P[∆ ≤ ❞❝♦❧ ]❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ate
d p
rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ate
d p
rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s ✇✐t❤ t❤❡✐r ♠❡❛♥✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ t❤r♦✇s✳
0 0.2 0.4 0.6 0.8 1 1.20
10
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50
60
70
80
Estimated probability (bin)
Nu
mb
er
of
po
ints
in
bin
Probability estimation histogram
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
0.2
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0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ate
d p
robabili
tyProbability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
10
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Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶ ✻ ✶✵ ✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶ ✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
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1.2x 10
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Simulation number
Estim
ate
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✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
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Estimated probability (bin)
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Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶.✻ · ✶✵−✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶.✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
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0.8
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1.2x 10
−5
Simulation number
Estim
ate
d p
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tyProbability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
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Estimated probability (bin)
Num
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oin
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Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶.✻ · ✶✵−✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶.✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1000
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1.2x 10
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Simulation number
Estim
ate
d p
robabili
tyProbability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
0 0.2 0.4 0.6 0.8 1 1.20
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Estimated probability (bin)
Num
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of p
oin
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Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✵✵✵✵ ❡①❛❝t❧② s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✶.✻ · ✶✵−✻✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✶.✼✷✺✽✳
❯♥r❡❧✐❛❜❧❡✦
◆❡❡❞ ❢♦r ❛ ✇❡❧❧ ❝❤♦s❡♥ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ t❡❝❤♥✐q✉❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❈▼❈ r❡s✉❧ts
❘❛r❡ ❡✈❡♥t t❡❝❤♥✐q✉❡ r❡q✉✐r❡♠❡♥ts
❆♣♣❧✐❝❛❜✐❧✐t②✿ ♠❡t❤♦❞ ♠✉st ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ♦♥ ❛ ❜❧❛❝❦ ❜♦①♠❛♣♣✐♥❣ ❛♥❞ ✇✐t❤♦✉t ♣r✐♦r✳
❆✛♦r❞❛❜✐❧✐t②✿ ♠❡t❤♦❞ s✐♠✉❧❛t✐♦♥ ❝♦st s❤♦✉❧❞ ❜❡ ❛s s♠❛❧❧ ❛s♣♦ss✐❜❧❡✳
❊♥❣✐♥❡❡r✲❢r✐❡♥❞❧✐♥❡ss✿ ♠❡t❤♦❞ s❤♦✉❧❞ ❜❡ ❛s s❡❧❢✲t✉♥✐♥❣ ❛s♣♦ss✐❜❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡❚❤❡ ❛❧❣♦r✐t❤♠❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts❈▼❈ ✈❡rs✉s ❆❙❚
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ ❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ ✳
❣ ❞ ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ ✳
❣ ❞ ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ ❞ ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
s ❳ ❣ ❤ ❳ ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
A ⊂ R ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❳ s ✶ ❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
A ⊂ R ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
P[ξ ∈ A] = P[❳ ∈ s−✶(A)]❄
■❢ ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ t②♣✐❝❛❧ ✐♥❞✉str✐❛❧ ♣r♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ♣r♦❜❧❡♠
❤ : X → R❞ ✐s t❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ❛♥❞ ❝❛♥ ❜❡ ❛ ❜❧❛❝❦ ❜♦①✳
❳ ∼ ❢❳ ✐s t❤❡ ✐♥♣✉t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♦♥ X✳
❣ : R❞ → R ✐s t❤❡ ❝r✐t❡r✐♦♥ ❢✉♥❝t✐♦♥✳
ξ ≡ s(❳ ) = ❣ ◦ ❤(❳ ) ✐s t❤❡ ♦❜s❡r✈❡❞ r❛♥❞♦♠ s❝♦r❡✳
A ⊂ R ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
P[ξ ∈ A] = P[❳ ∈ s−✶(A)]❄
■❢ P[ξ ∈ A] ✐s s♠❛❧❧✱ ❈r✉❞❡ ▼♦♥t❡ ❈❛r❧♦ ✇♦♥✬t ❞♦✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧
✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
✵
✐ ✶ ✐ ✐ ✶
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ ❚
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
{
R = A✵ ⊃ · · · ⊃ Aκ = AP[ξ ∈ A] =
∏κ✐=✶
P[ξ ∈ A✐ |ξ ∈ A✐−✶]
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ ❚
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
{
R = A✵ ⊃ · · · ⊃ Aκ = AP[ξ ∈ A] =
∏κ✐=✶
P[ξ ∈ A✐ |ξ ∈ A✐−✶]
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ ❚
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐♥ ❛ ♥✉ts❤❡❧❧✶ ❘❡♣❤r❛s❡ t❤❡ s♦✉❣❤t ♣r♦❜❛❜✐❧✐t②
{
R = A✵ ⊃ · · · ⊃ Aκ = AP[ξ ∈ A] =
∏κ✐=✶
P[ξ ∈ A✐ |ξ ∈ A✐−✶]
✷ ❊st✐♠❛t❡ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♣r♦❜❛❜✐❧✐t✐❡s ✐t❡r❛t✐✈❡❧②✳
❋r♦♠ ♥♦✇ ♦♥✱ s✉♣♣♦s❡ A = [❚ ,∞[
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ ❳ ✐
❦ ✶♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦ ✶✳
✷ ❉❡✜♥❡ ❦ ❛s s ❳ ✐
❦ ✶✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡t
❦ ❦ ✳✸ s ❳ ❦ ❦ ✶ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ ❦ ❛s s ❳ ✐
❦ ✶✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡t
❦ ❦ ✳✸ s ❳ ❦ ❦ ✶ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Generate ξi−1
points according to fξ|ξ≥ d
i−1
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ s ❳ ❦ ❦ ✶ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Quantile positioning
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳
✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ ❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥
❦ ✳✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡
❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Killing under the threshold
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦
✳✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i−1
Resampling above the threshold uniformly with replacement
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦
✳
✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i
Sampling conditionally to being above the threshold
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦
✳✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i
New quantile positionning
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡st❡♣ ❜② st❡♣
✶ ●❡♥❡r❛t❡ {❳ ✐
❦−✶} ♣♦✐♥ts
❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦−✶✳
✷ ❉❡✜♥❡ δ❦ ❛s {s(❳ ✐
❦−✶)}✬s
❡♠♣✐r✐❝❛❧ ❛❦ ✲q✉❛♥t✐❧❡ ❛♥❞ s❡tA❦ = [δ❦ ,∞[✳
✸ P[s(❳ ) ≥ δ❦ |A❦−✶] ∼ ❛❦ ✳✹ ❘❡♣❧❛❝❡ ♣♦✐♥ts ♦✉ts✐❞❡ A❦
❜② ❝❤♦♦s✐♥❣ ✉♥✐❢♦r♠❧② ✇✐t❤r❡♣❧❛❝❡♠❡♥t ❛♠♦♥❣ t❤♦s❡ ✐♥A❦ ✳
✺ ❯s❡ ❛❧❧ t❤❡ ♣♦✐♥ts t♦ s❛♠♣❧❡❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦
✳✻ ❇❛❝❦ t♦ st❡♣ ✷✳
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡✐♥ ✐♠❛❣❡s
0 200 400 600 800 10000
0.5
1
1.5
2
2.5
3
3.5
4
Generation number
ξ i
New quantile positionning
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❆❙❚ ❧♦❣✐❣r❛♠
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
"Use" points to generate
conditionally to being
above the quantile.
EndCalculate
sought probability
NO
YES
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❛❧❣♦r✐t❤♠
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
(Repeat ad libitum)
EndCalculate
sought probability
NO
YES
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼ ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
① ▼ ① ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
① ② ❢❳ ① ▼ ① ② ❢❳ ② ▼ ② ①
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
① ② ❢❳ ① ▼ ① ② ❢❳ ② ▼ ② ①
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
▼(·, ·) : X × X → R ✐s ♠❛♣♣✐♥❣ s✉❝❤ t❤❛t
∀① ∈ X, ▼(① , ·) : X → R ✐s ❛ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
▼ ✐s s❛✐❞ t♦ ❜❡ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ✐❢
∀(① , ②) ∈ X × X, ❢❳ (①)▼(① , ②) = ❢❳ (②)▼(② , ①)
▼❡❛♥✐♥❣✱ ✐❢ ❢r♦♠ ❛ ❢❳ s❡t✱ ②♦✉ ✉s❡ ▼ t♦ ❣❡♥❡r❛t❡ ❛♥♦t❤❡r✱
t❤❡ ♥❡✇ s❡t ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ ❛s ✇❡❧❧✳
st❛t✐st✐❝❛❧❧②✱ ♥♦ ♦♥❡ ❝❛♥ s❛② ✇❤✐❝❤ s❡t ❣❡♥❡r❛t❡❞ t❤❡ ♦t❤❡r✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ❢❳ s ❳ ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼ ❳❦ ✳
✷ ❳❦
❩ ✐❢ s ❩ ❦
❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼ ❳❦ ✳
✷ ❳❦
❩ ✐❢ s ❩ ❦
❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳❦
❩ ✐❢ s ❩ ❦
❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳❦ ▼❦ ❳❦ ❞② ✇❤❡r❡
▼❦ ❳❦ ❞② ✶❦s ② ▼ ❳❦ ❞② ▼ ❳❦
❝
❦ ❳❦❞②
❦ ❦
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳̃❦ ∼ ▼❦(❳❦ , ❞②) ✇❤❡r❡
▼❦(❳❦ , ❞②) = ✶A❦(s(②))▼(❳❦ , ❞②) + ▼(❳❦ ,A❝
❦)δ❳❦
(❞②)A❦ = [δ❦ ,+∞[
❳❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ s ❳ ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳̃❦ ∼ ▼❦(❳❦ , ❞②) ✇❤❡r❡
▼❦(❳❦ , ❞②) = ✶A❦(s(②))▼(❳❦ , ❞②) + ▼(❳❦ ,A❝
❦)δ❳❦
(❞②)A❦ = [δ❦ ,+∞[
❳̃❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦❛s✇❡❧❧✦
❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ s ❳ ❦s❡t✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❙❛♠♣❧✐♥❣ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦✉s✐♥❣ ❛ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼✳
●✐✈❡♥ ❳❦ ∼ ❢❳ |s(❳ )≥δ❦✱ s❡t
✶ ❩ ❛❝❝♦r❞✐♥❣ t♦ ▼(❳❦ , ·)✳
✷ ❳̃❦ =
{
❩ ✐❢ s(❩ ) ≥ δ❦❳❦ ♦t❤❡r✇✐s❡
✐✳❡✳ ❣✐✈❡♥ ❳❦ ✱ ❳̃❦ ∼ ▼❦(❳❦ , ❞②) ✇❤❡r❡
▼❦(❳❦ , ❞②) = ✶A❦(s(②))▼(❳❦ , ❞②) + ▼(❳❦ ,A❝
❦)δ❳❦
(❞②)A❦ = [δ❦ ,+∞[
❳̃❦ ✐s ❞✐str✐❜✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ ❢❳ |s(❳ )≥δ❦❛s✇❡❧❧✦
⇒ ❖♥❡ ❝❛♥ ✉s❡ ▼ t♦ ✐♥✢❛t❡ ❛ ❢❳ |s(❳ )≥δ❦s❡t✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆ ❝❧♦s❡r ❧♦♦❦ ❛t ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
(Repeat ad libitum)
EndCalculate
sought probability
NO
YES
❆ ❣❧♦❜❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❆❙❚ ✇✐t❤ ♠❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s t♠ ✐ r♠ ✐ ✈♠ ✐ ✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ ♠ ✐ ♠ ✐ ✵✻ t ✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ ✶✻❤✸✵ ✶✻❤✹✺ ✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝st ■ ❘✐ t t t♠ ✐ r♠ ✐ ✈♠ ✐ ♠ ✐ ♠ ✐ ✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ♠✐♥t ■
❘✶ t ❘✷ t
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡
■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❚▲❊s (t♠,✐ ,~r♠,✐ , ~✈♠,✐ )✳
●❛✉ss✐❛♥ ❛❞❞✐t✐✈❡ ✐✳✐✳❞✳ ♥♦✐s❡ (~ρ♠,✐ , ~ν♠,✐ ) ∼ N (✵✻,ΣtΣ)✳
❙t✉❞✐❡❞ t✐♠❡ ✐♥t❡r✈❛❧ ■ = [✶✻❤✸✵, ✶✻❤✹✺]✳
❑❡♣❧❡r✐❛♥ ❞❡t❡r♠✐♥✐st✐❝ ❞②♥❛♠✐❝s∀t ∈ ■ , ~❘✐ (t) = ~φ(t, t♠,✐ , (~r♠,✐ , ~✈♠,✐ ) + (~ρ♠,✐ , ~ν♠,✐ ))✳
▼✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ t✐♠❡ ✐♥t❡r✈❛❧
∆ (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) = ♠✐♥t∈■
{‖~❘✶(t) − ~❘✷(t)‖}
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ ♠ ✶ ♠ ✶ ♠ ✷ ♠ ✷ ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
❳ ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
✵ ❞❝♦❧ ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ = (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
❳ ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
✵ ❞❝♦❧ ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ = (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
ξ = ∆(❳ ) ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
✵ ❞❝♦❧ ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
◆♦t❛t✐♦♥s ❛♥❞ ◆♦✐s❡ ■■
❳ = (~ρ♠,✶, ~ν♠,✶, ~ρ♠,✷, ~ν♠,✷) ✐s t❤❡ ●❛✉ss✐❛♥ ✐♥♣✉t✳
ξ = ∆(❳ ) ✐s t❤❡ r❛♥❞♦♠ ♠✐♥✐♠✉♠ ❞✐st❛♥❝❡ ❞✉r✐♥❣ ■ ✳
A = [✵, ❞❝♦❧ ] ✐s t❤❡ t❛r❣❡t s❝♦r❡ s❡t✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points
●❡♥❡r❛t❡ r❛♥❞♦♠ ♥♦✐s❡s(
❳ ✶
❦, · · · ,❳ ♥
❦
)
✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❈❛❧❝✉❧❛t❡ ■r✐❞✐✉♠✲❑♦s♠♦s ❞✐st❛♥❝❡s ♦✈❡r t✐♠❡ ✐♥t❡r✈❛❧❧❡ ■ ✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❋✐♥❞ ♠✐♥✐♠❛(
∆✶
❦, · · · , ∆♥
❦
)
✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❋✐♥❞ ♠✐♥✐♠❛(
∆✶
❦, · · · , ∆♥
❦
)
✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❋✐♥❞ ♠✐♥✐♠❛(
∆✶
❦, · · · , ∆♥
❦
)
✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
(10) (20) (30) (40) (50) (60) (70) (80) (90) (100)200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❙♦rt ♠✐♥✐♠❛ t♦ ❢♦r♠(
∆(✶)❦
, · · · ,∆(♥)❦
)
✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
(10) (20) (30) (40) (50) (60) (70) (80) (90) (100)200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉❡✜♥❡ q✉❛♥t✐❧❡ δ❦+✶✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
(10) (20) (30) (40) (50) (60) (70) (80) (90) (100)200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
10 20 30 40 50 60 70 80 90 100200
400
600
800
1000
1200
1400
1600
Trial Number
Min
imal D
ista
nce
Sorted Random Minimal Distance
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
16.5 16.55 16.6 16.65 16.7 16.75
1000
2000
3000
4000
5000
6000
7000
Heure (h)
Dis
tance (
m)
Random Iridium−Kosmos distances
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points after discarding
❉✐s❝❛r❞ ♣♦✐♥ts ♦✈❡r q✉❛♥t✐❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❚❤❡ ❆❙❚ ✐♥ ❛❝t✐♦♥
❆♣♣❧✐❡❞ ❆❙❚ ✐♥ ✐♠❛❣❡s
Input points after shaking
❯s❡ t❤❡ ♠❛r❦♦✈ ❦❡r♥❡❧ ❛♥❞ s❡❧❡❝t❡❞ ♣♦✐♥ts t♦ r❡❣❡♥❡r❛t❡ s❛♠♣❧❡✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
rob
ab
ility
Probability estimation
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s ✇✐t❤ t❤❡✐r ♠❡❛♥✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳❞❝♦❧ = ✶✵✵♠✱✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) t❤r♦✇s✳
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Nu
mb
er
of
po
ints
in
bin
Probability estimation histogram
❋✐❣✉r❡✿ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
robabili
ty
Probability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✹ ✼✽ ✶✵ ✼✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵ ✸✷✸✷✳
◗✉✐t❡ ❘❡❧✐❛❜❧❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
robabili
ty
Probability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✹.✼✽ · ✶✵−✼✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵.✸✷✸✷✳
◗✉✐t❡ ❘❡❧✐❛❜❧❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ✈✐❛ ❆❙❚ r❡s✉❧ts
❆❙❚ ❡st✐♠❛t✐♦♥✳ ❞❝♦❧ = ✶✵✵♠
0 10 20 30 40 50 60 70 80 90 1001
2
3
4
5
6
7
8
9
10x 10
−7
Simulation number
Estim
ate
d p
robabili
ty
Probability estimation
✭❛✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥s
1 2 3 4 5 6 7 8 9 10
x 10−7
0
5
10
15
20
25
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
✭❜✮ Pr♦❜❛❜✐❧✐t② ❡st✐♠❛t✐♦♥ ❤✐st♦❣r❛♠
❋✐❣✉r❡✿ ✶✵✵ ❡st✐♠❛t✐♦♥s ❜❛s❡❞ ♦♥ ✸✵✾✵✻✵(✶± ✷%) s✐♠✉❧❛t✐♦♥s✳
▼❡❛♥ ❡st✐♠❛t❡✿ ✹.✼✽ · ✶✵−✼✳
❊♠♣✐r✐❝❛❧ r❡❧❛t✐✈❡ ❞❡✈✐❛t✐♦♥✿ ✵.✸✷✸✷✳
◗✉✐t❡ ❘❡❧✐❛❜❧❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❊st✐♠❛t✐♦♥s
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ate
d p
robability
Probability estimation
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❊st✐♠❛t✐♦♥s ❛♥❞ ♠❡❛♥s
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ate
d p
robability
Probability estimation
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❍✐st♦❣r❛♠s
0 0.2 0.4 0.6 0.8 1 1.20
10
20
30
40
50
60
70
80
Estimated probability (bin)
Num
ber
of p
oin
ts in b
in
Probability estimation histogram
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈▼❈ ✈❡rs✉s ❆❙❚
❈▼❈ ❱s ❆❙❚
❋✐❣✉r❡✿ ❊st✐♠❛t✐♦♥s
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
1.2x 10
−5
Simulation number
Estim
ated p
robab
ility
Probability estimation
❋✐❣✉r❡✿ ❍✐st♦❣r❛♠s
0 0.2 0.4 0.6 0.8 1 1.20
10
20
30
40
50
60
70
80
Estimated probability (bin)
Numb
er of
point
s in bi
n
Probability estimation histogram
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s❖✈❡r✈✐❡✇❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧◗✉❛♥t✐❧❡ ❧❡✈❡❧
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
(Repeat ad libitum)
EndCalculate
sought probability
NO
YES
❆ ❣❧♦❜❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡ ❆❙❚ ✇✐t❤ ♠❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
EndCalculate
sought probability
NO
YES
❍♦✇❡✈❡r✱ ♥♦ r❛t✐♦♥❛❧ q✉❛♥t✐❧❡ ❧❡✈❡❧ ❝❤♦✐❝❡✳✳✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
EndCalculate
sought probability
NO
YES
✳✳✳♥♦r ▼❛r❦♦✈✐❛♥ ❝❤♦✐❝❡ ❛♥❞ t✉♥✐♥❣✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❦❡② ❞✐✣❝✉❧t✐❡s
❆❙❚ ❧♦❣✐❣r❛♠ ✇✐t❤ ▼❛r❦♦✈✐❛♥ r❡s❛♠♣❧✐♥❣ ❛♥❞ ❞✐✣❝✉❧t✐❡s
Start
Generate according to the original
law.
k=0
Calculate empirical quantile
Calculate conditional probability
k=k+1
Replace points under quantile with points above,
choosing uniformly with replacement.
Empirical quantile over
threshold?
Propose transitions with the
Reversible Markov Kernel.
Accept those above the quantile.
Refuse the others
EndCalculate
sought probability
NO
YES
❚✇♦ ✐ss✉❡s t♦ t❛❝❦❧❡✦
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❚❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣ ❚❡❝❤♥✐q✉❡ ✐s
❛ ✐♥t❡r❛❝t✐♥❣ ♣❛rt✐❝❧❡ t❡❝❤♥✐q✉❡ ❬✷❪
❛ ✈❛r✐❛♥❝❡ r❡❞✉❝t✐♦♥ ♠❡t❤♦❞
✉s❡❞ ❛s ❛ r❛r❡ ❡✈❡♥t ❞❡❞✐❝❛t❡❞ ▼♦♥t❡ ❈❛r❧♦ ♠❡t❤♦❞ ❬✺✱ ✻❪✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❇❛s✐❝ ■♥❣r❡❞✐❡♥ts ❬✶❪
●✐✈❡♥
❢❳ ■♥♣✉t Pr♦❜❛❜✐❧✐t② ❉❡♥s✐t② ❋✉♥❝t✐♦♥❚ ❚❤r❡s❤♦❧❞ t♦ ❡①❝❡❡❞◆ ❙✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❑♥♦✇♥ ♦r ❞❡s✐❣♥❡❞
▼ ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❉❡❝✐❞❡❞ ❛♥❞ ❞❡t❡r♠✐♥✐st✐❝
❛❦ ❦t❤ ◗✉❛♥t✐❧❡ ❧❡✈❡❧❜❦ ❦t❤ ❦❡r♥❡❧ ♣❛r❛♠❡t❡r s❡t♥❦ ❦t❤ st❡♣ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❖❜s❡r✈❡❞ ❛♥❞ r❛♥❞♦♠
❦ ❦t❤ ❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡❚♦t❛❧ ♥✉♠❜❡r ♦❢ q✉❛♥t✐❧❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❖✈❡r✈✐❡✇
❇❛s✐❝ ■♥❣r❡❞✐❡♥ts ❬✶❪
●✐✈❡♥
❢❳ ■♥♣✉t Pr♦❜❛❜✐❧✐t② ❉❡♥s✐t② ❋✉♥❝t✐♦♥❚ ❚❤r❡s❤♦❧❞ t♦ ❡①❝❡❡❞◆ ❙✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❑♥♦✇♥ ♦r ❞❡s✐❣♥❡❞
▼(·, ·) ❢❳ ✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❉❡❝✐❞❡❞ ❛♥❞ ❞❡t❡r♠✐♥✐st✐❝
❛❦ ❦t❤ ◗✉❛♥t✐❧❡ ❧❡✈❡❧❜❦ ❦t❤ ❦❡r♥❡❧ ♣❛r❛♠❡t❡r s❡t♥❦ ❦t❤ st❡♣ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t
❖❜s❡r✈❡❞ ❛♥❞ r❛♥❞♦♠
δ❦ ❦t❤ ❡♠♣✐r✐❝❛❧ q✉❛♥t✐❧❡κ ❚♦t❛❧ ♥✉♠❜❡r ♦❢ q✉❛♥t✐❧❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ✵❞ ■❞ ✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼ ① ❞②①
✶ ❜✷
❜✷
✶ ❜✷
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧②
❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
●❛✉ss✐❛♥✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧ ❬✶❪
■❢ ❳ ∼ N (✵❞ , ■❞ )✱ t❤❡♥ ❛ ♣♦ss✐❜❧❡ ❦❡r♥❡❧ ❝❤♦✐❝❡ ✐s
▼(① , ❞②) ∼ N(
①√✶ + ❜✷
,❜✷
✶ + ❜✷
)
❯♣ t♦ ♥♦✇✱ ♣❛r❛♠❡t❡r ❜ ✐s ✐♥✐t✐❛t❡❞ ❤❡✉r✐st✐❝❛❧❧② ❛♥❞ ♦♣t✐♦♥❛❧❧② s❡❧❢t✉♥❡s ❛s ❛ ✇❤♦❧❡ s❡t ❞❡♣❡♥❞❡♥t ▼❛r❦♦✈ ❝❤❛✐♥✳
●❡♥❡r❛❧ ❝❛s❡
■♥ ♦t❤❡r ❝❛s❡s✱ ♥♦ ♦✛✲t❤❡✲s❤❡❧❢ r❡✈❡rs✐❜❧❡ ❦❡r♥❡❧✳
▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛❧❣♦r✐t❤♠ ❝❛♥ ❤❡❧♣ ❬✽❪✳
❆ ✇✐❞❡r ❛rr❛② ♦❢ ❞❡♥s✐t②✲r❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❦❡r♥❡❧ ♣❛✐r ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦ ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳
❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦ ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳
■t❡r❛t✐♥❣ ▼❦ ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦(·, ·) ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
❚❤❡ ❘❡✈❡rs✐❜❧❡ ▼❛r❦♦✈ ❑❡r♥❡❧
❙❛♠♣❧❡ ❞❡♣❡♥❞❡♥❝❡
❙❛♠♣❧❡s ❛r❡ ❝♦rr❡❧❛t❡❞ ❛♥❞ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛s t❤❡② s❤❛r❡❛ ❝♦♠♠♦♥ ❣❡♥❡❛❧♦❣②✳ ❚❤✐s ❝♦rr❡❧❛t✐♦♥ ❝❛♥ r❡s✉❧t ✐♥t♦ ✈❛r✐❛♥❝❡✳■t❡r❛t✐♥❣ ▼❦(·, ·) ♥❡✈❡r ❤✉rts ❛♥❞ ❝❛♥ r❡❞✉❝❡ ✈❛r✐❛♥❝❡ ❬✽✱ ✼❪✳
❍♦✇ t♦ ❝r❡❛t❡ ✐♥❞❡♣❡♥❞❡♥❝❡❄❍♦✇ ❢❛st ✐s ✐♥❞❡♣❡♥❞❡♥❝❡ r❡❛❝❤❡❞❄
❍♦✇ t♦ ❝r❡❛t❡ ❞✐✈❡rs✐t②❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ✷✵ ✷✺ ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ∈ [.✷✵, .✷✺] ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ∈ [.✷✵, .✷✺] ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
◗✉❛♥t✐❧❡ ❧❡✈❡❧
▼♦st ❡✣❝✐❡♥t ❝♦♠❜✐♥❛t✐♦♥ ♦❢ q✉❛♥t✐❧❡ ❧❡✈❡❧s
❆❝❝♦r❞✐♥❣ t♦ ❬✹❪✱ t♦ ♠✐♥✐♠✐s❡ ✈❛r✐❛♥❝❡✱ ✉♥❞❡r ♠✐❧❞ ❛ss✉♠♣t✐♦♥s✱ ❛❧❧q✉❛♥t✐❧❡ ❧❡✈❡❧s ❛❦ s❤♦✉❧❞ ❜❡ ❡q✉❛❧✱ s❛② t♦ ❛✳
❇❡st ❧❡✈❡❧ ❝❤♦✐❝❡
❚❤❡r❡ ✐s ♥♦ ✐❞❡❛ ♥♦✇ ♦❢ ❛♥ ♦♣t✐♠❛❧ ❝❤♦✐❝❡ ❢♦r ❛ ✦
❊①tr❡♠❡❧② ❤✐❣❤ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ❛✈♦✐❞❡❞ ❛s ✇❡ ✇❛♥t t♦ ❛✈♦✐❞r❛r❡ ❡✈❡♥ts✳
❊①♣❡r✐♠❡♥t❛❧❧②✱ ❛ ∈ [.✷✵, .✷✺] ✇♦r❦s ✜♥❡ ❬✶❪✳
❱❡r② ❧♦✇ ❛ t♦♦✱ ✇❤❡♥ ❞❡❛❧t ✇✐t❤ ❝❛r❡❢✉❧❧❧② ❬✸❪✳
❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡❞✉❝❡ ✈❛r✐❛♥❝❡❄❍♦✇ t♦ ❝❤♦♦s❡ ❛ t♦ r❡s♣❡❝t t❤❡ s✐♠✉❧❛t✐♦♥ ❜✉❞❣❡t❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❑❡② ✐❞❡❛s t♦ r❡♠❡♠❜❡r
❆❙❚ ✇✐❞❡❧② ♦✉t♣❡r❢♦r♠s ❈▼❈ ✇❤❡♥ ✐t ❝♦♠❡s t♦ r❛r❡ ❡✈❡♥ts ✳
❘✉❧❡ ♦❢ t❤❡ t❤✉♠❜ t✉♥✐♥❣ ❝❛♥ ❞♦ t❤❡ tr✐❝❦✳✳✳
✳✳✳❜✉t t❤❡♦r❡t✐❝❛❧ ✉♥❞❡rst❛♥❞✐♥❣ ✐s ♥❡❡❞❡❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ ❳ ♣ ♣ t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ ❳ ♣ ♣ t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ P [❳ ∈ A] ∈ [♣−, ♣+] t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❋✉rt❤❡r ✇♦r❦✿ t❤❡♦r❡t✐❝❛❧ ♠❛st❡r②
❆♣♣❧②✐♥❣ ✐t t♦ ❛ ✇✐❞❡r ❝❧❛ss ♦❢ r❛♥❞♦♠ ✐♥♣✉t ✈✐❛▼❡tr♦♣♦❧✐s✲❍❛st✐♥❣ ❛♥❞ ✉s✐♥❣ ❡♠♣✐r✐❝❛❧ st♦♣♣✐♥❣ ❝r✐t❡r✐♦♥✳
❯s✐♥❣ ❡①tr❛ ❦♥♦✇❧❡❞❣❡ P [❳ ∈ A] ∈ [♣−, ♣+] t♦ ❝❤♦♦s❡♣❛r❛♠❡t❡rs✳
❙❤♦✉❧❞ ✇❡ r❡s❛♠♣❧❡ ❛❧❧ t❤❡ ♣♦✐♥ts ♦r ♦♥❧② t❤❡ ❞♦✉❜❧♦♦♥❄
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
P❧❡❛s❡✱ ❧❡t ♠❡ ❛♥s✇❡r ②♦✉r q✉❡st✐♦♥s✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❈♦♥❝❧✉s✐♦♥
❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
❈♦♥❢❡r❡♥❝❡s
✶ ❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ✈❡rs✉s ❞❡❜r✐s ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡✿ ✽t❤ ✐♥t❡r♥❛t✐♦♥❛❧ ✇♦r❦s❤♦♣ ♦♥ r❛r❡
❡✈❡♥t s✐♠✉❧❛t✐♦♥✱ P♦st❡r✱ ✷✵✲✷✸ ❏✉♥❡ ✷✵✶✵ ❈❛♠❜r✐❞❣❡ ✭❯❑✮✱ ❘✳P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✷ ❘❡♣r❡s❡♥t✐♥❣ ❙♣❛t✐❛❧ ❉✐str✐❜✉t✐♦♥ ✈✐❛ t❤❡ ❆❞❛♣t✐✈❡ ❙♣❧✐tt✐♥❣t❡❝❤♥✐q✉❡ ❛♥❞ ■s♦q✉❛♥t✐❧❡ ❈✉r✈❡s ✿ ❊✉r♦♣❡❛♥ ▼❡❡t✐♥❣ ♦❢
❙t❛t✐st✐❝✐❛♥s ✷✵✶✵✱ Prés❡♥t❛t✐♦♥ ❖r❛❧❡ s✉r rés✉♠é✱ ✶✼✲✷✷ ❆✉❣✉st✷✵✶✵ P✐r❛❡✉s ✭●r❡❡❝❡✮✱ ❘✳ P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✸ ❋r♦♠ s❛t❡❧❧✐t❡ ✈❡rs✉s ❞❡❜r✐s ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s t♦ t❤❡ ❛❞❛♣t✐✈❡s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡ ❋r♦♠ ❆♣♣❧✐❝❛t✐♦♥ t♦ ❚❤❡♦r② ✿ ❘❛r❡ ❊✈❡♥t
❙✐♠✉❧❛t✐♦♥ ❲♦r❦s❤♦♣✱ Prés❡♥t❛t✐♦♥ ❖r❛❧❡ s✉r rés✉♠é✱ ✷✽✲✷✾❖❝t♦❜❡r ✷✵✶✵ ❇♦r❞❡❛✉① ✭❋r❛♥❝❡✮✱ ❘✳ P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✹ ❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣t❡❝❤♥✐q✉❡ ✿ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ✐♥ ❈♦♠♣✉t❡r ▼♦❞❡❧✐♥❣ ❛♥❞
❙✐♠✉❧❛t✐♦♥✱ Prés❡♥t❛t✐♦♥ ❖r❛❧❡ s✉r ❛rt✐❝❧❡✱ ✼✲✾ ❏❛♥✉❛r② ✷✵✶✶▼✉♠❜❛✐ ✭■♥❞✐❛✮✱ ❘✳ P❛st❡❧✱ ❏✳ ▼♦r✐♦ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
❘❡✈✉❡s
✶ ❙❛♠♣❧✐♥❣ ❚❡❝❤♥✐q✉❡ ❢♦r ❧❛✉♥❝❤❡r ✐♠♣❛❝t s❛❢❡t② ③♦♥❡❡st✐♠❛t✐♦♥ ✿ ❆❝t❛ ❛str♦♥❛✉t✐❝❛✱ ✻✻✭✺✲✻✮✿ ✼✸✻✲✼✹✶✱ ✷✵✶✵✱ ❏✳▼♦r✐♦ ✫ ❘✳ P❛st❡❧✳
✷ ❆♥ ♦✈❡r✈✐❡✇ ♦r ✐♠♣♦rt❛♥❝❡ s♣❧✐tt✐♥❣ ❢♦r r❛r❡ ❡✈❡♥t s✐♠✉❧❛t✐♦♥ ✿❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ P❤②s✐❝s✱ ✸✶✿✶✷✾✺✲✶✸✵✸✱ ✷✵✶✵✱ ❏✳ ▼♦r✐♦✱ ❘✳P❛st❡❧ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
✸ ❊st✐♠❛t✐♦♥ ❉❡ ♣r♦❜❛❜✐❧✐tés ❡t ❞❡ q✉❛♥t✐❧❡s r❛r❡s ♣♦✉r ❧❛❝❛r❛❝tér✐s❛t✐♦♥ ❞✬✉♥❡ ③♦♥❡ ❞❡ r❡t♦♠❜é❡ ❞✬✉♥ ❡♥❣✐♥ ✿ ❏♦✉r♥❛❧
❞❡ ❧❛ ❙♦❝✐été ❋r❛♥ç❛✐s❡ ❞❡ ❙t❛t✐st✐q✉❡✱ ♣❡♥❞✐♥❣ s✉❜❥❡❝t t♦♠♦❞✐✜❝❛t✐♦♥s✱ ❏✳ ▼♦r✐♦✱ ❘✳ P❛st❡❧ ✫ ❋✳ ▲❡ ●❧❛♥❞✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❖✉t❧✐♥❡
✶ ■♥tr♦❞✉❝t✐♦♥
✷ ■r✐❞✐✉♠ ❛♥❞ ❈♦s♠♦s
✸ ❆♥ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
✹ ❆❙❚✬s ♦♣❡♥ ✐ss✉❡s
✺ ❈♦♥❝❧✉s✐♦♥
✻ P✉❜❧✐❝❛t✐♦♥s ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥s
✼ ❘❡❢❡r❡♥❝❡s
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❬✶❪ ❋ré❞ér✐❝ ❈ér♦✉✱ P✐❡rr❡ ❉❡❧ ▼♦r❛❧✱ ❚❡❞❞② ❋✉r♦♥✱ ❛♥❞ ❆r♥❛✉❞●✉②❛❞❡r✳❘❛r❡ ❡✈❡♥t s✐♠✉❧❛t✐♦♥ ❢♦r ❛ st❛t✐❝ ❞✐str✐❜✉t✐♦♥✳❚❡❝❤♥✐❝❛❧ r❡♣♦rt✱❤tt♣✿✴✴❤❛❧✳✐♥r✐❛✳❢r✴✐♥r✐❛✲✵✵✸✺✵✼✻✷✴❢r✴✱ ✷✵✵✾✳
❬✷❪ P✐❡rr❡ ❉❡❧ ▼❖❘❆▲✳❋❡②♥♠❛♥✲❑❛❝ ❋♦r♠✉❧❛❡✿ ●❡♥❡❛❧♦❣✐❝❛❧ ❛♥❞ ■♥t❡r❛❝t✐♥❣ P❛rt✐❝❧❡
❙②st❡♠s ❲✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✳❙♣r✐♥❣❡r✱ ✷✵✵✹✳
❬✸❪ ❆r♥❛✉❞ ●✉②❛❞❡r✱ ◆✐❝♦❧❛s ❲✳ ❍❡♥❣❛rt♥❡r✱ ❛♥❞ ❊r✐❝▼❛t③♥❡r✲▲ø❜❡r✳■t❡r❛t✐✈❡ ▼♦♥t❡ ❈❛r❧♦ ❢♦r ❡①tr❡♠❡ q✉❛♥t✐❧❡s ❛♥❞ ❡①tr❡♠❡♣r♦❜❛❜✐❧✐t✐❡s✳❚❡❝❤♥✐❝❛❧ r❡♣♦rt✱ ▼❛r❝❤ ✷✵✶✵✳P❛♣❡r ❛♥❞ s❧✐❞❡s❤♦✇ ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❛✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡✳
❬✹❪ ❆❣♥és ▲❛❣♥♦✉①✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❘❛r❡ ❡✈❡♥t s✐♠✉❧❛t✐♦♥✳P❊■❙✱ ✷✵✭✶✮✿✹✺✕✻✻✱ ❏❛♥✉❛r② ✷✵✵✻✳❆✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡✳
❬✺❪ P✐❡rr❡ ▲✬➱❝✉②❡r✱ ❋r❛♥ç♦✐s ▲❡ ●❧❛♥❞✱ P❛s❝❛❧ ▲❡③❛✉❞✱ ❛♥❞ ❇r✉♥♦❚✉✣♥✳❙♣❧✐tt✐♥❣ ♠❡t❤♦❞s✳■♥ ●❡r❛r❞♦ ❘✉❜✐♥♦ ❛♥❞ ❇r✉♥♦ ❚✉✣♥✱ ❡❞✐t♦rs✱ ▼♦♥t❡ ❈❛r❧♦
▼❡t❤♦❞s ❢♦r ❘❛r❡ ❊✈❡♥t ❆♥❛❧②s✐s✱ ❝❤❛♣t❡r ✸✱ ♣❛❣❡s ✸✾✕✻✶✳ ❏♦❤♥❲✐❧❡② ✫ ❙♦♥s✱ ❈❤✐❝❤❡st❡r✱ ✷✵✵✾✳
❬✻❪ ❋r❛♥ç♦✐s ▲❊●▲❆◆❉✳❋✐❧tr❛❣❡ ❜❛②és✐❡♥ ♦♣t✐♠❛❧ ❡t ❛♣♣r♦①✐♠❛t✐♦♥ ♣❛rt✐❝✉❧❛✐r❡✳▲❡s Pr❡ss❡s ❞❡ ❧✬❊❝♦❧❡ ◆❛t✐♦♥❛❧❡ ❙✉♣ér✐❡✉r❡ ❞❡ ❚❡❝❤♥✐q✉❡s❆✈❛♥❝é❡s✱ ✷✵✵✽✳❆✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡ ✐♥ ❋r❡♥❝❤✳
❬✼❪ ❙✳ P✳ ▼❡②♥ ❛♥❞ ❘✳ ▲✳ ❚✇❡❡❞✐❡✳▼❛r❦♦✇ ❝❤❛✐♥s ❛♥❞ st♦❝❤❛st✐❝ st❛❜✐❧✐t②✳❙♣r✐♥❣❡r✕❱❡r❧❛❣✱ ✶✾✾✸✳
❊st✐♠❛t✐♥❣ s❛t❡❧❧✐t❡ ❝♦❧❧✐s✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ✈✐❛ t❤❡ ❛❞❛♣t✐✈❡ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡
❘❡❢❡r❡♥❝❡s
❬✽❪ ▲✉❦❡ ❚■❊❘◆❊❨✳▼❛r❦♦✈ ❝❤❛✐♥s ❢♦r ❡①♣❧♦r✐♥❣ ♣♦st❡r✐♦r ❞✐str✐❜✉t✐♦♥s✳❆♥♥❛❧s ♦❢ ❙t❛t✐st✐❝s✱ ✷✷✿✶✼✵✶✕✶✼✷✽✱ ❉❡❝❡♠❜❡r ✶✾✾✹✳❆✈❛✐❧❛❜❧❡ ♦♥ ❧✐♥❡✳