Computable randomness is inherently imprecise Gert de Cooman and - - PowerPoint PPT Presentation

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Computable randomness is inherently imprecise Gert de Cooman and - - PowerPoint PPT Presentation

Jul 11, 2023 245 likes •501 views

Computable randomness is inherently imprecise Gert de Cooman and Jasper De Bock ISIPTA 2017 Lugano, 10 July 2017 A single forecast A single forecast A single forecast Gambles available to Sceptic: interval forecast f ( 0 ) f ( 1 ) f ( 0 )

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Computable randomness is inherently imprecise

Gert de Cooman and Jasper De Bock ISIPTA 2017 Lugano, 10 July 2017

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A single forecast

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A single forecast

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A single forecast

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E

p

( f ) = Ep(f) = 0 f(1) f(0) f(1) ≤ f(0) f(1) ≥ f(0)

Gambles available to Sceptic: interval forecast

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Gambles available to Sceptic: interval forecast

f(1) f(0) f ( 1 ) ≤ f ( ) f ( 1 ) ≥ f ( ) E

r

( f ) =

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Gambles available to Sceptic: interval forecast

f(1) f(0) E0( f) = 0 E1(f) = 0 f(1) ≤ f(0) f(1) ≥ f(0)

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00 000 001 01 010 011 1 10 100 101 11 110 111

Event tree

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00 000 001 01 010 011 1 10 100 101 11 110 111 I⇤ I0 I1 I00 I11 I10 I01

Forecasting system

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00 000 001 01 010 011 1 10 100 101 11 110 111 I⇤ I0 I1 I00 I11 I10 I01

Computable randomness of a sequence

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00 000 001 01 010 011 1 10 100 101 11 110 111 I1 I2 I2 I3 I3 I3 I3

Consistency results

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00 000 001 01 010 011 1 10 100 101 11 110 111 I I I I I I I

Constant interval forecasts

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Church randomness

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The set filter of forecasts that make a sequence random

1 pC(ω) pC(ω)

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Gambles available to Sceptic: interval forecast

f(1) f(0) E0(f) = 0 E1( f) = 0 f(1) ≤ f(0) f(1) ≥ f(0)

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The set filter of forecasts that make a sequence random

1 pC(ω) pC(ω)

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Ep( f) = 0 E

p

( f ) = f(1) f(0) f ( 1 ) ≤ f ( ) f ( 1 ) ≥ f ( )

Gambles available to Sceptic: interval forecast

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The set filter of forecasts that make a sequence random

1 pC(ω) pC(ω)

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Interval randomness: a simple example

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Point randomness, but not quite

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And where do we go from here?

  • 1. Is it possible to use an equivalent Martin-Löf type

approach, using randomness tests?

  • 2. Can we take other notions of computability into

account?

  • 3. Are similar results possible on a prequential approach?
  • 4. Our results seem to allow for an ontological

interpretation of imprecise probabilities: how do we do statistics with them?

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And where do we go from here?

  • 1. Is it possible to use an equivalent Martin-Löf type

approach, using randomness tests?

  • 2. Can we take other notions of computability into

account?

  • 3. Are similar results possible on a prequential approach?
  • 4. Our results seem to allow for an ontological

interpretation of imprecise probabilities: how do we do statistics with them?

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And where do we go from here?

  • 1. Is it possible to use an equivalent Martin-Löf type

approach, using randomness tests?

  • 2. Can we take other notions of computability into

account?

  • 3. Are similar results possible on a prequential approach?
  • 4. Our results seem to allow for an ontological

interpretation of imprecise probabilities: how do we do statistics with them?

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And where do we go from here?

  • 1. Is it possible to use an equivalent Martin-Löf type

approach, using randomness tests?

  • 2. Can we take other notions of computability into

account?

  • 3. Are similar results possible on a prequential approach?
  • 4. Our results seem to allow for an ontological

interpretation of imprecise probabilities: how do we do statistics with them?