1. The relation between causality and probability Marianne Belis Ecole Supérieure d’Informatique, Paris

2. 2 Common features of causality and probability Both are models of change Both have predictive and explicative value Both have philosophical, scientific and practical aspects Both use empiric (unsatisfactory) definitions Both need consideration of the single case in order to find the ontological relation between them

3. 3 Singular causes Nancy Cartwright (1989) : - Singular causes are basic - Anti-humean view of causality - Repetitive singular processes lead to long range regular associations The problem of patterns of regularities - Econometrics: error terms in the causal relations - Probabilistic causality (Patrick Suppes) - Bayesian networks (Judea Pearl – AI) ………………. The resort to empirical probabilities overshadows the singular case

4. 4 The random causal process What are the objective features at the singular level How these features justify the patterns of regularity? What distinguishes random from deterministic processes at the singular level? Two concepts help answering these questions: - capacity - propensity

5. 5 Capacities Cartwright : individual properties give rise to capacities manifest themselves in interaction observable and measurables (scientific concept) depend upon the situation Example (Cartwright): aspirin carries the capacity to relieve headaches (because of its chemical compounds, aspirin has properties which carry the capacity to interact with the brain and release headaches; this interaction has a certain strength)

6. 6 Propensity – Aristotle: a potentiality that is inherent in the individual things - a general, universal property – Popper: a relational property of the experimental arrangement - propensities are responsible for singular events - can be measured by a potential (virtual) statistical frequency in the case of random repetitive processes

7. 7 Propensity vs. capacity A causal process includes many active entities: C 1, C 2,…, C n (usually named causes and conditions) Each active entity has a capacity to influence the effect The capacities have various strengths: c 1, c 2, …, c n The propensity of the effect e j to occur is due to all these influences: Prop(e j ) = F(c 1j, c 2j,…,c nj ) Simplified form (algebraic sum): Prop(e j ) = ∑c ij i = 1,…,n Positive strengths support, negative strengths weaken the occurrence of the effect

8. 8 Example: the growth of a plant Active entities (causes and conditions): C 1 =seed, C 2 = humidity, C 3 =warmth, C 4 =fertilizers Each of them has a capacity for making the plant to grow Strengths of the respective capacities: c 1, c 2, c 3, c 4 Outputs: e 1 = plant ; e 2 = no plant. The global force which brings the plant into life is Prop(e 1 ) = Σ i c i1 i = 1…4 Prop(e 1 ) is the propensity of the plant to grow.

9. 9 Constant and variable propensities Causal strengths can be constant or variable Constant strengths don’t vary during the process and its repetitions Variable strengths arise inside or outside the process (irregular variations with zero mean value) Constant propensity + irregular variations = instantaneous propensity (variable propensity greater or smaller than the constant one)

10. 10 Continuous and discontinuous behaviour - Continuous behaviour: small variations of the parameters of the cause lead to small variations of the parameters of the effect (same state) - Discontinuous behaviour: small variations of the parameters of the cause lead to qualitative changes of the effect (change of state) - The threshold of qualitative transformation (critical value of a parameter which entails a change of state)

11. 11 Examples of continuous/discontinuous behaviour Continuous rise of water  the dam bursts Continuous accumulation of snow  avalanche Continuous deposition of cholesterol  thrombosis Continuous rise of social discontent  revolution

12. 12 Objective features of randomness - Discontinuous link cause effect - Constant propensities with values close to the threshold of qualitative transformation - Irregular variations of the causal strengths leads to instantaneous propensities able to reach the threshold and to entail a change of state

13. 13 The objective features of “chance” A causal process with a number of mutual exclusive effects Each effect is supported by its own instantaneous propensity Instantaneous propensities compete for producing their effect The first which reaches the threshold wins the race The effect whose constant propensity is closer to the threshold has more facility to occur (with the aid of irregular variations) Chance is represented by the value of the constant propensity

14. 14 The role of irregular variations  They exist in all types of connections  Negligible in continuous types (deterministic) as they lead only to a variation of parameters  Important in discontinuous types (random) as they lead to a change of state  The observer knowledge doesn’t matter

15. 15 From propensities to probabilities Propensities are difficult to evaluate as capacities strengths are often hard to measure in absolute values An universal (dimensionless) measure of chance would require a relative value (a normalization of the absolute value): Relative propensity = the ratio of the absolute propensity to the total propensity involved in the process: Relative Prop(e j ) = ∑c ij / ∑ c ij i = 1,…,n; j = 1,..,m i i,j It is a proportion of causal strengths (proportion is the key concept of probability)

16. 16 Causal transmission channel The input signals (ct. propensities) generates output signals (outcomes) Constant propensities + noise  instantaneous propensities (variable) The one which reaches first the threshold give rise to an outcome Instantaneous propensities noise constant propensities (inputs) By repeating the transmission each output will occur a number of times proportional to its input (the one closest to the threshold will pass more often) outcomes (outputs)

17. 17 A stochastic analog to digital converter …transforms the input signals in numbers by eliminating the noise (zero mean value in the long run) For the causal transmission, in the long run: n j = k Prop(e j ) The total number of transmissions: N = n 1 + n 2 + ….+ n m = k( Prop (e 1 ) +k Prop (e 2 ) +…+ kProp(e m )) = k Prop(E) where E = {e 1, e 2,…, e m } is the set of outputs (effects) After a great number of transmissions: n j / N = Prop(e j ) / Prop(E) = probability(e j )

18. 18 Prior and posterior probabilities Normalized propensities = prior probabilities Frequency of outcomes = posterior probabilities causal transmission channel noise Laplace’s side: he evaluated the normalized propensities (prior probabilities) in a simple case: the urn with balls (equal causal strengths) von Mises’ side: he counted the relative outcomes (posterior probabilities)

19. 19 Conclusions The singular case is fundamental for both types of causal connections (deterministic or random) Capacities and propensities characterize the single case The objective probability of the single case is basic and it justifies the long range behaviour of random processes Probability is a ratio of constant causal strengths. This represents the ontological relation between causality and probability