The document discusses the complexities of interpreting statistical indicators and experimental data within the context of decision-making and reliability. It emphasizes the influence of sampling, human judgment, and the inherent uncertainties in deriving accurate values or trends from various sources. The conclusions suggest a need for careful consideration of the methodologies used in statistical predictions and the validation of results against benchmark data.

1. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Most take things upon trust
Bartolomeo Civalleri1
, Roberto Dovesi1
, Erin R. Johnson3
,
Pascal Pernot 4
, Davide Presti2
, Andreas Savin5
1
Department of Chemistry and NIS Centre of Excellence
University of Torino (Italy)
2
Department of Chemical and Geological Sciences and INSTM research unit
University of Modena and Reggio-Emilia, Modena (Italy)
3
Chemistry and Chemical Biology, School of Natural Sciences
University of California, Merced (USA)
4
Laboratoire de Chimie Physique d’Orsay
Universit´e Paris-Sud (France)
5
Laboratoire de Chimie Th´eorique
CNRS and Sorbonne University UPMC Univ Paris 6 (France)
Winterschool on Computational Chemistry 2015
1 / 91

2. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Most take things upon trust
... some (and those of the most) taking
things upon trust, misemploy their power of
assent ...
John Locke
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3. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Most take things upon trust
... some (and those of the most) taking
things upon trust, misemploy their power of
assent ...
John Locke
2 / 91

4. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Success of benchmarks
Quantifying experience
0
50
100
150
200
250
300
1993 1998 2003 2008 2013
0
2000
4000
6000
8000
10000
12000
Numberofpublications
Numberofcitations
Year
publications
citations
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5. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
What this talk is about: statistics
Dealing with a large amount of numbers:
efficient algorithms
performant computers
new methods, e.g., DFT
Statistical methods used to concentrate information
largely used in environmental sciences, medicine, finance, ....
very useful
pitfalls
In spite of mathematical rigor, using statistical indicators does not
avoid human decision.
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6. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Predicting with or without understanding
Physical models with systematic improvement:
understanding
Improvement can be seen with optimism
Limitations:
cost and time
absence of rigorous bounds
Statistical models (correlations):
without knowing the underlying cause
Legitimate when used with necessary care
Limitations:
Choice of quantities (properties) entering the model
Statistical treatment
Conclusions drawn
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7. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Overview
Properties (quantities) analyzed
Quality of approximation (model)
Decisions to take (human)
(How do the preceding points affect the design of methods?)
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8. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Unjustified correlations
Happiness(t) = w0 + w1
t
j=1
γ
t−j
CRj + w2
t
j=1
γ
t−j
EVj + w3
t
j=1
γ
t−j
RPEj
R.B. Rutledge et al, PNAS 2014
Was happiness properly
defined?
Were the factors
determining it properly
chosen?
How is the agreement of
the data with the model?
Do we learn how to get
happier?
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9. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Justified correlations: predicting without
understanding
Captain James Cook forced his crew to eat sauerkraut, without
knowing that lack of vitamin C produces scurvy.
Properties: sauerkraut (containing vitamin C) and number of
sailors getting scurvy
Agreement: very good (although no statistics)
Acting: Cook acted, and avoided scurvy
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10. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Justified or unjustified?
Comparison of data obtained
by a model (e.g., density functional approximation, DFA),
and reference values (experimental, or calculated by a more
advanced model)
20 15 10 5
20
15
10
5
reference
B3LYP
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11. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Overview
Properties we study
Measures of satisfaction (statistical indicators)
Decisions to take (human)
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12. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Origin of properties
Do we get properties we need
from experiment?
from calculations?
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13. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Reference data from experiment
Do we get properties from experimental data?
Error bars
Corrections
Models used to analyze the data
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14. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Error bars
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15. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Is 1 kcal/mol chemical accuracy?
Results for the G1 set (atomization energies for 55 molecules)
The experimental data reported here are taken from a combination of NIST-JANAF
tables and Huber and Herzberg, ...
Most experimental errors are small, i.e., < 0.5 kcal/mol, although several are somewhat
larger, e.g., CS has an experimental error of 6 kcal/mol. ..
For several species experimental errors are unavailable.
J.C. Grossman, 2002
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16. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Is 1 kcal/mol chemical accuracy?
Results for the G1 set (atomization energies for 55 molecules)
The experimental data reported here are taken from a combination of NIST-JANAF
tables and Huber and Herzberg, ...
Most experimental errors are small, i.e., < 0.5 kcal/mol, although several are somewhat
larger, e.g., CS has an experimental error of 6 kcal/mol. ..
For several species experimental errors are unavailable.
J.C. Grossman, 2002
Difficulty to extract experimental error bars from published data
Cf. J. Cioslowski et al, 2000
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17. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Corrections to experimental data
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18. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Temperature dependence of lattice constants
Lattice constants measured to 5 significant digits
How many digits at 0 K? Hebstein, Acta Cryst B 2000
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19. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Models behind experimental data
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20. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Fundamental band gaps
R. T. Poole, J. G. Jenkin, J. Liesegang, R. C. Leckey
Phys. Rev. B 11, 5679, 1975
Independent particle
model
Origin of data
PES, and inverse
PES?
exciton structure?
. . .
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21. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Spurious effects on experimental data?
For example, Taylor and Hartman tentatively placed the valence
band of LiF at about 13 eV below the vacuum level on the basis of
an edge in the photoelectric yield curve. However, their yield curve
continues to fall rapidly at lower photon energies, and this may be
interpreted as a threshold of approximately 10 eV, which compares
favorably with our estimate of 9.8 eV for this quantity.
R. T. Poole, J. G. Jenkin, J. Liesegang, R. C. Leckey Phys. Rev. B 11, 5679, 1975
Problems for band gaps: nuclear motion, surface, ... effects, etc.
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22. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Ideal and real experimental data
ZnO Weinen et al. (report from cpfs.mpg.de)
Reproduce the spectrum not the gap
(H. Tjeng, in Lausanne 2014)
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23. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Reference data from calculations
Same quantity can be calculated with reference and model
Is the theory behind the calculation capable to provide the
desired quantity?
Can we trust calculated data?
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24. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Is the theory behind the calculation capable to
provide the desired quantity?
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25. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Calculating fundamental band gaps with different
methods
IP − EA
Provided by exact Green’s functions
Not provided by Hartree-Fock∗
Not provided by exact Kohn-Sham orbital energies
1966: Sham-Kohn, 1983: Perdew-Levy, Sham-Schl¨uter, ...∗
Exact Kohn-Sham calculations would provide exact results using two
separate calculations, for X, and for X−
Density functional hybrids∗
?
∗
Just correlation for most calculations?
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26. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Do we use the right theory?
smartandgreen.eu
Do we need fundamental or optical band gaps?
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27. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Can we trust calculated data?
Higher quality calculations may not
have the accuracy needed for comparisons with lower level
methods
be allowed to “filter” (to decide whether a lower level
calculation is good or bad)
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28. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Accuracy of “reliable” calculations
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29. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
How accessible are sufficiently accurate data?
Mean absolute errors (kcal/mol)
G1 set (atomization energies for 55 molecules)
J.C. Grossman, 2002
B3LYP DMC CCSD(T)/aug-cc-pVQZ CCSD(T)/CBS
2.5 2.9 2.8 1.3
27 / 91

30. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
How accessible are sufficiently accurate data?
Mean absolute errors (kcal/mol)
G1 set (atomization energies for 55 molecules)
J.C. Grossman, 2002
B3LYP DMC CCSD(T)/aug-cc-pVQZ CCSD(T)/CBS
2.5 2.9 2.8 1.3
Does the reference have the same (in)accuracy?
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31. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Effect of improving the calculated reference data
Weak interaction benchmark data set (S22)
given set of molecules
2006 calculations (Jurecka et al.)
2011 calculations (Marshall et al.)
Percentage of “correct” results changes from 55% to 86%
“Correct”: within ±0.5 kcal/mol with B3LYP corrected for dispersion by XDM
20 15 10 5
20
15
10
5
reference
B3LYP
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32. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Filtering using “reliable” calculations
Perform reference calculations with a different method, and refrain
from accepting results when the two methods disagree.
Example: Perform point-wise an expensive calculation to verify
cheap calculation.
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33. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Why filtering not necessarily is an improvement
Cases:
Model says Filter selects Filter rejects
result reliable a (correct) b (error)
result unreliable c (error) d (correct)
Fraction of reliable results:
before selection by filter: (a + b)/(a + b + c + d)
after selection by filter: a/(a + c)
Filtering brings no improvement when
(a + b)/(a + b + c + d) ≥ a/(a + c), or bc ≥ ad
30 / 91

34. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Filters not always successful
Band gaps in 28 cubic crystals
Filter is too restrictive, but reliable
PBEsol PBE filter selects (39%) PBE filter rejects (61%)
Result reliable (93%) 11 15
Result unreliable(7%) 0 2
100% (11 out of 11) results selected by PBE filter are reliable
Filter selects reasonably well, and is useless
PBEsol PBE0 filter selects (92%) PBE0 filter rejects (8%)
Result reliable (93%) 24 2
Result unreliable(7%) 2 0
Results selected by PBE0 filter are reliable, ≈ as without filter, but systems excluded.
PBEsol and PBE0 make the same mistake: unreliable results are close, and thus wrongly selected (ad = 0). Some reliable
PBEsol results are not close to PBE0, and rejected (bc = 0). Thus, ad < bc.
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35. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Reference data. Summary
Pitfalls:
Experimental data
Error bars
Corrections applied
Model used
Calculated data
may not be accurate enough
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36. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Benchmarks
Model and benchmark?
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37. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Benchmarks
Benchmark and reality
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38. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Reference data. Conclusion
To judge the quality of a method we compare to benchmarks.
These can be inappropriate.
Need for critical analysis of the accuracy of the reference data
from the perspective of the problem under study.
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39. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Reference data. Conclusion
To judge the quality of a method we compare to benchmarks.
These can be inappropriate.
Need for critical analysis of the accuracy of the reference data
from the perspective of the problem under study.
Once we have decided about reference data, we have to
define a measure quantifying our choice: statistical indicators
34 / 91

40. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Overview
Properties we study
Measures of satisfaction (statistical indicators)
Decisions to take (human)
35 / 91

41. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Diagnostic tools
Weighing of the heart Book of the Dead, Papyrus of Ani, British Museum
With large amount of numbers: need for representative numbers
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42. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Statistical indicators. Overview
Many indicators (mean, median, mode, ...)
Role of sampling
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43. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Many indicators
Indicators can yield different ranking
When the mean has no meaning
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44. Trust
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Introduction
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Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Indicators can yield different ranking
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45. Trust
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Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Indicators do not yield the same ordering of
methods
Absolute errors: |xi |
Mean: 1
n i=1,n |xi |
Median: half of the |xi | are < median, half > median
Maximum: max(|x1|, |x2|, . . . , |xN |)
Results for the G3/99 benchmark set (kcal/mol)
Method Mean Median Max
B3LYP 4 2 34
LC-ωPBE 5 4 25
Which method is better?
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46. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Condensing data by indicators
Radiation around Fukushima Daiichi NPS
NNSA 04/03/2011
Evacuation at 30 km, exclusion zone 20 km
Radiation may more important at >30 km than at 10 km.
Mean: bad indicator
41 / 91

47. Trust
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Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Origin of problems
Error distribution, and its mean
mean is relevant irrelevant
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48. Trust
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Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
A simple model for parametrized approximations
A mathematically (=clearly) defined problem
Model Analogy
x ∈ (0, 1) Choice of system (random)
(1 + x)2
Exact result
1 + mx Approximation
m ∈ (2, 3) Parameter
y = (1 + mx) − (1 + x)2
Error of the approximation
Objective: “Recommend the best m”
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49. Trust
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Indicators can yield
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When the mean has
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Domain of validity
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Psychology of
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Publishing reliable
results
Conclusions
A set of simple models
m = 2 using exact value of function and derivative at origin
m = 3 using exact value of function at origin and endpoint
2 < m < 3 using exact value of function at origin, and some
other criterion of similitude
44 / 91

50. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
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uncertainties
Accurate values or
trends
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results
Conclusions
Approximations do not yield normally distributed
errors
Model and its error distribution
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51. Trust
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Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Absolute error distributions
Origin of difference between medians (red), max, mode, . . .
B3LYP LC wPBE
count
0204060
count
0 20 40 60
0
10
20
30
absolute error
Two density functional methods, for G3/99
46 / 91

52. Trust
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
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uncertainties
Accurate values or
trends
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Utility
Psychology of
decision
Publishing reliable
results
Conclusions
When the mean has no meaning
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53. Trust
A. Savin
Introduction
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Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
When the mean has no meaning
h
1
Α
α is uniformly distributed on (0, π/2).
What is the mean value of h?
48 / 91

54. Trust
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Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
When the mean has no meaning
h
1
Α
α is uniformly distributed on (0, π/2).
What is the mean value of h?
π/2
0
dα tan(α) =
∞
0
h d arctan(h) =
∞
0
h dh
1+h2 = ∞
Variance also diverges.
cf. Lorentzian shape for peaks in spectroscopy
48 / 91

55. Trust
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Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
No mean, no variance, ...?
B3LYP distribution of atomization energy errors (G3/99)
Distributed as 2
πa (1 + (h/a)2
)−1
?
0 5 10 15 20 25 30 35
0
50
100
150
200
absolute error kcal mol
frequency
B3LYP
Mean on sample: 4 kcal/mol
Explanation: small errors accumulate in larger systems
49 / 91

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When the mean has
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Indicators are
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uncertainties
Accurate values or
trends
Domain of validity
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Psychology of
decision
Publishing reliable
results
Conclusions
Shape of distributions
Nearly uniform distribution of PBE absolute errors, G3/99
0 10 20 30 40 50
0
50
100
150
absolute error kcal mol
frequency
PBE
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57. Trust
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Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Small errors accumulate in larger systems. A
simple model
Error of approximation when detaching an atom from a chain with
n bonds (n + 1 atoms) ≈ x
x
Error in the atomization energy: x n
Mean error for chains of n = 1, . . . , m:
1
m
m
n=1
x n =
1
m
x m(m + 1)/2 = x(m + 1)/2
diverges when m → ∞
The error of the atomization energy per atom → x, when m → ∞.
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Introduction
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Small errors accumulate in larger systems.
Atomization energies
G3, G2, and G1 benchmark sets with different functionals
MAE MAE/atom
G1
G2
G3
B3LYP
CAM-B3LYP
LC-ωPBE
B97-1
BLYP
PBE0
PW86PBE
PBE
BH&HLYP
G1
G2
G3
B3LYP
CAM B3LYP
LC ΩPBE
B97 1
BLYP
PBE0
PW86PBE
PBE
BH&HLYP
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59. Trust
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Indicators are affected by sampling
53 / 91

60. Trust
A. Savin
Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Uncertainty of mean
54 / 91

61. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Finite sampling brings uncertainty of mean
Simple example
Uniform sampling on interval (0,1):
Distribution of 100 means of samples of 100
0.45 0.50 0.55
0
5
10
15
mean
count
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62. Trust
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Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Finite sampling brings uncertainty of mean
G3/99 MAEs
(and subsets with randomly reduced sample size - from 221 to 22)
Full G3 99 set Subset 1 Subset 2 Subset 3
1
2
3
4
5
6
MAE kcal mol
B97 1
CAM B3LYP
B3LYP
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63. Trust
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Indicators can yield
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When the mean has
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Indicators are
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Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Finite sampling brings uncertainty of mean
Benchmarks for weak interactions
0.2
0.3
0.4
0.5
0.6
B3LYP
BLYP
LC
−ωPBE
PW
86PBEBH
&H
LYPC
AM
−B3LYPPBE0
B97−1
PBE
MeanAbsoluteError(kcal/mol)
KB49
S22
S66
S115
Are differences between methods important?
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64. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Statistical indicators. Summary
Different statistical indicators can lead to different
conclusions, and may even not exist
Sampling is unavailable, and brings uncertainty
58 / 91

65. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Composite Portraiture
59 / 91

66. Trust
A. Savin
Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Statistical indicators. Conclusions
Statistical indicators (MAE, etc.)
are useful, maybe unavoidable, but
can induce into error, and thus
should be used with care
In spite of mathematical formulation,
supplementary criteria needed
60 / 91

67. Trust
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Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Overview
Properties we study
Measures of satisfaction (statistical indicators)
Decisions to take (human)
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68. Trust
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Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Actions (decisions) after knowing the results of
the (statistical) analysis
Living with uncertainties
Accurate values or accurate trends
Domain of validity
Utility
Psychology of decision
Publishing only correct results
62 / 91

69. Trust
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Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Living with uncertainties
63 / 91

70. Trust
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Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Uncertainties in reference values affect judgment
of calculated data
Lattice constants in some cubic crystals: MSE±RMSD
LDA: -3.5±2.7 pm
HSEsol (best among tested functionals): 0.0±1.5 pm
Is the source of the error in the reference data?
Is there a need for better functionals?
64 / 91

71. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
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Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Uncertainty in reference data propagates to
ranking of functionals
What is agreement with experimental data, if we do not know how
accurate experimental data are?
Percentage of computed (dispersion corrected) results
in agreement with experimental (G3/99) atomization energies,
within ±x kcal/mol
Method x = 0.5 x = 1 x = 2 x = 4
BLYP 9 14 24 42
B3LYP 9 22 44 73
CAMB3LYP 7 13 29 57
65 / 91

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different ranking
When the mean has
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Accurate values or
trends
Domain of validity
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Conclusions
Accurate values or trends?
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73. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Accurate values or trends?
Which method is better?
0
Good mean (more accurate) Good variance (better trend)
MAE not a good indicator: mixes systematic with random errors
66 / 91

74. Trust
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Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Mean errors and variance for band gap
calculations
2 0 2 4 6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
ME
Σ
HF LDA PBE PBEsol
PBE0 PBEsol0 B97 B3LYP
HSE06 HSEsol HISS LC wPBE
LC wPBEsol RSHXLDA wB97 wB97 X
Prefer HISS?
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75. Trust
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Indicators can yield
different ranking
When the mean has
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Living with
uncertainties
Accurate values or
trends
Domain of validity
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Psychology of
decision
Publishing reliable
results
Conclusions
Correcting for the mean error of band gaps
Constant shift easy to correct (new approximation):
error → error-ME (choice by σ)
2 0 2 4 6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
ME
Σ
HF LDA PBE PBEsol
PBE0 PBEsol0 B97 B3LYP
HSE06 HSEsol HISS LC wPBE
LC wPBEsol RSHXLDA wB97 wB97 X
Prefer LC-ωPBEsol?
With one parameter HF can be made as good as HISS.
68 / 91

76. Trust
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
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affected by sampling
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uncertainties
Accurate values or
trends
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decision
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results
Conclusions
Correcting lattice constants
69 / 91

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When the mean has
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uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Domain of validity
70 / 91

78. Trust
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Introduction
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Relevance of the benchmark data set
Questions
Is the benchmark relevant for the problem of interest?
E.g., not when benchmark designed for one property, and used for another
Is the benchmark biased? The benchmark is based upon
systems that may be different from the system under study,
e.g., because of a shift of interest with time
Example: Do not expect good large band gaps, based upon the experience with
small band gaps
0 2 4 6 8 10 12 14
0
1
2
3
0
1
2
3
band gap eV
bandgapeV
HSE06
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different ranking
When the mean has
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affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Which method is better?
Band gaps of a set of crystals: better use HSE06 or HISS?
0 2 4 6 8 10 12
0.0
0.5
1.0
1.5
band gap eV
absoluteerroreV
HSE06
HISS
72 / 91

80. Trust
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Avoiding mixing?
Not always possible: systems where
different “components” are simultaneously needed,
the relative importance of “components” is not reproduced
the same way by different methods.
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
When mixing produces inversions
We split the S115 benchmark set into:
1. H-bonded (HB)
2. weakly interacting (WI)
One of the method gives better MAEs for both benchmark sets.
Situations where the worse method gives better results?
74 / 91

82. Trust
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
When mixing produces inversions
We split the S115 benchmark set into:
1. H-bonded (HB)
2. weakly interacting (WI)
One of the method gives better MAEs for both benchmark sets.
Situations where the worse method gives better results?
Yes, when weighing of HB and WI is different!
HB WI
4
5
6
7
8
9
10
Interaction
MAE
B3LYP
PBE0
74 / 91

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Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Best method not for all properties
Mean absolute errors
Method Lattice constant (pm) Bulk modulus (GPa)
HSEsol 1 7
PBE0 4 5
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different ranking
When the mean has
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Domain of validity
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Publishing reliable
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Utility
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85. Trust
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From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Utility of a calculation
Choose between A and B
Result A (expensive) B (cheap)
A better than B good bad
A as good as B good good
A worse than B bad good
77 / 91

86. Trust
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Introduction
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Properties
From experiment
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Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Quantify the utility of a calculation
“Gain” obtained by choosing between A and B
cheap: +1, expensive, -1; good: +2, bad, -2
A (expensive) B (cheap)
A better than B good (+2-1) bad (-2+1)
A as good as B good (+2-1) good (+2+1)
A worse than B bad (-2-1) good (+2+1)
78 / 91

87. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Quantify the utility of a calculation for lattice
constants
Choice between A (PBEsol) and B (LDA) for lattice constants
A (expensive) B (cheap) probability
A better than B 1 -1 1/2
A as good as B 1 3 1/4
A worse than B -3 3 1/4
“Gain” 0 1
MAE 2.4 pm 3.5 pm
MAE (or probability) and “gain” give contradictory recommendations
79 / 91

88. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Psychology of decision
Market behavior
80 / 91

89. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
What is the best method?
Safest result?
Minimize maximal error
Most stable error on average?
81 / 91

90. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Share portfolio
Dow Jones Industrial Average on 2 June 2014
Company Yearly change
Pfitzer Inc -3.26
Walt Disney Co 9.96
http://money.cnn.com/data/markets/dow
Share portfolio does not insure highest gain, but enhances stability
Supposing long-term gain at stock exchange
82 / 91

91. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Using a portfolio of methods?
Using different methods, in the right proportion can enhance
stability of results
Do 25% of calculations with RPA and 75% with RPAx
HB7
WI8
Type of interaction
RPA
RPAx
Method
2.0
2.5
3.0
Error
Data from W. Zhu et al, JCP 2010
83 / 91

92. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Using a portfolio of methods?
Using different methods, in the right proportion can enhance
stability of results
Do 25% of calculations with RPA and 75% with RPAx
HB7
WI8
Type of interaction
RPA
RPAx
Method
2.0
2.5
3.0
Error
Data from W. Zhu et al, JCP 2010
Mixing by community
Invisible hand of the market
83 / 91

93. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
The probability to predict the correct result
Assumptions:
New calculations yield acceptable results, distributed as in the
benchmark set.
The probability to obtain a good result is given by
p =
a
t
a: number of results within the accepted threshold,
t: total number of results in the benchmark set
Probability to obtain at least k correct answers, out of 10
(binomial distribution)
10
j=k
10
j
pj
(1 − p)10−j
84 / 91

94. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Probability to publish n correct results out of ten
The probability to obtain B3LYP+XDM atomization energies with
absolute errors per atom less than a chosen maximum acceptable
value, for a set of n systems and assuming the same error
distribution as the G3 data set.B3LYP errors, based upon G3/99
atomization energies/atom
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Probability
Maximum acceptable absolute error (kcal/mol)
10
5
2
1
Even with a very large tolerance (3 kcal/mol per atom) it is more probable to not have all
ten results right, than to have them all right.
85 / 91

95. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Human decisions. Summary
Not necessarily the same choice, if made for best accuracy, or
for best trend
The community’s choice (experience with one class of
systems, properties, ...) might not be the best adapted to the
problem under study
Criteria from decision theory can orient choices otherwise
than “diagnostic tools” (like MAE, etc.)
Statistics tell us that there are many unreliable results in the
literature
86 / 91

96. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Human decision. Conclusions
Decision taking is unavoidable.
Specifying the criteria for the decision taken should be part of
the study.
87 / 91

97. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Conclusions
What should I do ...?
Auguste Rodin Le Penseur Mus´ee Rodin
88 / 91

98. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Conclusions
Many pitfalls. Recommendations
Learning statistics and decision theory is useful when working
with large amounts of data
89 / 91

99. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Conclusions
Many pitfalls. Recommendations
Learning statistics and decision theory is useful when working
with large amounts of data
Good old effort for understanding should not be forgotten
35th Midwest Theoretical Chemistry Conference, Ames (2003)
Speaker: I use DFT, because it is an easy to use black box, and
does not require much thinking.
K. Ruedenberg: Why is it a bad thing to think?
Calculations get easy, but expertize still needed
89 / 91

100. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Some (difficult) questions to try to answer
What do we want to know from the calculation?
Is the theory we use capable to provide it?
What accuracy do we need?
Do we expect the approximations we make give the necessary
accuracy?
On what is based our judgment (knowledge, experience,
advice, impact factor, ...)?
Are the reference data significant for our problem?
How do we judge the accuracy of the approximation
(sufficient data, significant indicators, . . . ) ?
Is their accuracy sufficient for our purpose?
If the accuracy is not sufficient, what are we willing to give
up?
. . .
90 / 91

101. Trust
A. Savin
Introduction
Overview
Properties
From experiment
From calculations
Statistical indicators
Many indicators
Indicators can yield
different ranking
When the mean has
no meaning
Indicators are
affected by sampling
Human decisions
Living with
uncertainties
Accurate values or
trends
Domain of validity
Utility
Psychology of
decision
Publishing reliable
results
Conclusions
Progress in science does not necessarily come
from better accuracy
Nearly uniform distribution of BH&HLYP absolute errors, G3/99
0 10 20 30 40 50
0
50
100
150
absolute error kcal mol
frequency
BH&HLYP
BH&H: at the origin of hybrids
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