1. 1 Dealing with uncertainty in international migration predictions: From probabilistic forecasting to decision analysis Jakub Bijak Division of Social Statistics and Centre for Population Change University of Southampton Joint Eurostat / UNECE Work Session on Demographic Projections Lisbon, 28–30 April 2010

2. 2 Contents Uncertainty in international migration studies Shift of perspective in demographic forecasting Further step: decision support –Bayesian decision analysis –Stylised migration example Limitations of migration predictions Outlook for the future

3. 3 Uncertain International Migration Uncertainty of measurement –Incoherent definitions, lack of harmonisation –Incomplete measurement of events/transitions Uncertainty of determinants –Lack of a good, comprehensive migration theory –Determinants and their impact on migration also largely uncertain (economy? individual decisions?) Uncertainty of the future (immanent) –Migration processes very difficult to predict, likely exhibiting non-stationary features

4. 4 Uncertain International Migration All these uncertainties cumulate in migration (and thus also population) forecasts One of key assumptions concerns stationarity Implications: –Computationally convenient, uncertainty “under control” –The effects of shocks, such as policy interventions, are expected to fade out over time –But: “too orderly” forecasts that tell more about current politics and society than migration [Pijpers, 2008] –“Too precise” predictions very likely to fail

5. 5 Perspective Shift in Forecasting Point forecasts / projections –Almost certainly will NOT come true Variant forecasts / projections –What probabilities are associated with the scenarios? Probabilistic forecasts –How to quantify the uncertainty?  Frequentist vs. Bayesian approaches –Which trajectory should the users choose?  No universal solution

6. 6 Perspective Shift in Forecasting A Bayesian example of migration forecasts Source: Bijak and Wiśniowski (2010) JRSS A, 174(4), forthcoming Which trajectories to choose for the decision making? Immigration to Portugal: 50%, 80% and 90% predictive intervals

7. 7 Bayesian Decision Analysis Bayesian decision analysis applies to various types of problems, such as: –Estimation and measurement –Forecasting Choice of an optimal decision depends on: –The underlying probability distribution –The decision setting, approximated by the cost of under- or overestimation or prediction

8. 8 Bayesian Optimal Decisions Rule: Choose a decision d that minimises the expected loss under a given probability distribution of states of the world w, p(w) Expected loss, under loss function L(w,d): ρ(p, d) = Solution (if exists): d* = arg min {d} { ρ(p, d) } [DeGroot, 1970]

9. 9 Loss Functions Loss function L(w,d) measures decision maker’s cost of making a decision d, when the true state of the world is w [DeGroot, 1970] –Symmetric functions L(w, d) = a (w – d) 2 ⇒ Solution: mean L(w, d) = a |w – d| ⇒ Solution: median L(w, d) = a ∙ 1 w≠d ⇒ Solution: mode –Asymmetric functions, more likely in real- world applications [Lawrence et al., 2006]

10. 10 Loss Functions: Examples Asymmetric loss functions [e.g. Zellner, 1986] –LinLin (linear-linear): L(w, d) = a (w – d) ∙ 1 w ≤ d + b (d – w) ∙ 1 w > d Solution: quantile of rank b/(a+b) –LinEx (linear-exponential): L(w, d) = b {exp[a (w – d)] – a (w – d) – 1} Solution: (–1/a) ln{E w [exp(–aw)]} Related to the moment-generating function

11. 11 Loss Functions: Examples Source: own elaboration

12. 12 Example: Stylised forecast Assume that a forecast of log(immigration) was prepared using a simple Bayesian linear model with Normal errors –For precision (inverse variance) of the error term, assume a priori Gamma distribution –For other parameters, assume a priori Normal distributions (standard assumptions) Predictive distribution for log(immigration) is Student’s t, and for immigration: log-t

13. 13 Example: Loss functions Problem: log-t distribution is heavy tailed and does not have positive moments Plausible loss functions: –LinLin (solutions: quantiles) –Point (solution: mode) Implausible loss functions: –Quadratic (mean does not exist) –LinEx (moment generating function ­ ditto) ☑☒☑☒

14. 14 Example: Decisions Solutions: 54,071 59,874 51,620 69,448 Source: own elaboration

15. 15 Limitations of forecasting Bayesian decisions Loss function: linear Loss function: non-linear Light-tailed distributions Solutions based on quantiles Some solutions usually exist Heavy-tailed distributions Solutions likely do not exist [See also: Taleb, 2009] Common-sense strategies

16. 16 Limitations of forecasting Bayesian decisions Loss function: linear Loss function: non-linear Light-tailed distributions Solutions based on quantiles Some solutions usually exist Heavy-tailed distributions Solutions likely do not exist [See also: Taleb, 2009] Migration predictions

17. 17 Alternatives and extensions Alternative: minimax rule –Decision minimising the maximum loss –But, how do we know what is the maximum? –Besides, very pessimistic… Bayesian alternative: conditional  -minimax –Minimax rule, but constrained to a certain class  of prior distributions –Extensions: stable decisions, which minimise the oscillations of risk over  [Męczasrki 1998]

18. 18 Limitations and caveats Forecasts cannot be too precise to avoid the “illusion of control” [Makridakis & Taleb, 2009] –E.g. the 2003 predictions of A8 immigration into the UK, off by over an order of magnitude Plausible forecast horizon vs stationarity: –Migration: 5 to 10 years? [cf. Holzer, 1959] –Population: up to 20 years? [cf. Keyfitz, 1982] Longer periods: back to scenarios? [Keilman, 1990]

19. 19 The Future Research on complexity and predictability of migration and population processes Towards more interactive forecasting? –Forecast users → provide the decision setting –Domain experts → provide information a priori –Forecasters → combine information to provide forecasts and user-specific decision support Common framework: Bayesian approach?

20. 20 Thank you! Research prepared within the Centre for Population Change (CPC) funded by the ESRC Grant number RES-625-28-0001. All opinions are those of the author only.