Normic support and the revision of prior knowledge

In three previous posts I discussed Martin Smith's paper "Why throwing 92 heads in a row is not surprising" [1]. I attempted a Bayesian interpretation of the concept of surprise, but I was sure that this had already been done before; a cursory literature search confirmed this impression (see below). Can one go further? Martin relates surprise to the more general concept of normic support, and the obvious question is whether the latter can also be interpreted in Bayesian terms.

I'll use the example of judicial evidence, that Martin treats in Ref. [2], where normic support is defined as follows :

a body of evidence E normically supports a proposition P just in case the circumstance in which E is true and P is false would be less normal, in the sense of requiring more explanation, than the circumstance in which E and P are both true.

The Blue Bus paradox can then be solved by arguing that testimonial evidence has normic support, while statistical evidence does not ([2], page 19). To put it in the terms above, finding out that the testimonial evidence is false would surprise us, while the failing of statistical evidence would not.

Can we restate this idea in terms of belief update, as I tried to do for surprise and, in particular, is the distinction between testimonial and statistical evidence similar to that between the coin throw and the lottery examples I drew here? The proposition P being in both cases "the bus involved was a Blue-Bus bus", we need to identify the evidence E of each type.

1. testimonial: the witness can identify the color of the bus with 90% accuracy.
2. statistical:   90% of the buses operating in the area on the day in question were Blue-Bus buses.

By quick analogy with the respective coin throw and the lottery examples, respectively, we can then say:

1. If the witness is wrong, the result
• calls into question his/her priorly presumed accuracy and prompts us to revise our estimate (Bayesian interpretation.)
• surprises us and requires more explanation (normic support perspective.)
2. If the bus is not blue then, although non-blue buses only account for 10% of the total,
• since the Blue-Bus deduction was merely based on the proportion of each type of bus there is no prior knowledge to revise (Bayesian interpretation.)
• the result is unlikely but not abnormal, and thus it does not call for further explanation (normic support perspective.)

I'll discuss in future posts how similar the two interpretations are and whether they solve the paradox (right now my feeling is that they don't, but I need to think about it some more.)

Bayesian surprise

A reference on the Bayesian treatment of surprise, defined as the Kullback-Leibler divergence of the posterior distribution with respect to the prior one.

Itti, L., & Baldi, P. F. (2006). Bayesian surprise attracts human attention. In Advances in neural information processing systems (p. 547–554).

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<sup>1. Martin Smith, Why throwing 92 heads in a row is not surprising, Philosophers' Imprint (forthcoming) 2017.↩
2. Martin Smith, When does evidence suffice for conviction?, Mind (forthcoming.)↩</sup>

Some choices are not surprising

In two previous posts I discussed Martin Smith's paper "Why throwing 92 heads in a row is not surprising" [1].

I argued that the all-heads sequence is more surprising than a more balanced one (composed of roughly equal numbers of heads and tails), although the two have the same probability of occurrence based on the prior information (fair and independent coins), because the first event challenges this information, while the second does not.

Prompted by an email exchange with Martin (whom I thank again for his patient and detailed replies!) I would like to discuss here cases where I believe no particular outcomes would be surprising. Let us take an example from the same paper, the lottery. I agree with the author that a draw consisting of consecutive numbers (e.g. 123456) is not surprising, nor is any other pattern or apparently random sequence, which all have the same probability of occurrence.

In my view, this is simply because –for the lottery case– equiprobability assumption is just an uninformative prior. Finding a patterned sequence does not challenge our conviction, because there is nothing to challenge: any outcome will do equally well. For the coin toss, on the other hand, the equiprobability of all sequences stems from the very strong conviction that the coins are (1) unbiased and (2) independent, and the all-head outcome does challenge it.

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^{1. Martin Smith, Why throwing 92 heads in a row is not surprising, Philosophers' Imprint (forthcoming) 2017.↩}

Impressions from ECIS 2017 - day 2

Highlights from the morning session of the second day. I managed to miss Jacob Klein's plenary talk (on Interfacial water).

Parallel session on topics 5 and 6 (roughly, inorganic colloids)

• Andrés Guerrero-Martínez (Madrid University) on the reshaping, fragmetation and welding of gold nanoparticles using femtosecond lasers.
• Two more talks on responsive Au@polymer systems: Jonas Schubert (Dresden University) and Rafael Contreras-Cáceres (Málaga University)
• Pavel Yazhgur (postdoc at the ESPCI, Paris after a remarkable PhD at the LPS, Orsay!) on hyperuniform binary mixtures. I should write a post on hyperuniformity at some point...

Parallel session on topic 3 (polymers, liquid crystals and gels)

• Hans Juergen Butt on the crystallization of polymers or water in alumina pores.

Retiro

I went jogging in the Buen Retiro park this evening. It reminds me of the Parc de la Tête d'Or, where I used to run many years ago. The difference is that back then I would overtake pretty much everybody. Nowadays, it's the other way round.

Impressions from ECIS 2017

I'm in Madrid for the 31st conference of the European Colloid and Interface Society. Here are some highlights from the morning session:

Michael Cates on active colloids (plenary session)

I arrived late and missed some of this talk, plus I'm not a specialist in the area of active colloids. What I found interesting is the search for the minimal modification of the various Hohenberg-Halperin models (B and H) that yield interesting behaviour; I still haven't understood how breaking the time-reversal symmetry comes into play. Here is a reference I promised myself I would read on the flight back home.

Parallel session on topics 5 and 6 (roughly, inorganic colloids)

Two talks on secondary structures in gold nanoparticle systems with potential applications to SERS:

• Christian Kuttner (Dresden University) discussed the protein-mediated formation of raspberry-shaped gold clusters consisting of small particles adsorbed onto larger ones. Nice SAXS measurements reveal the presence of both particle populations.
• Joachim Koetz (Potsdam University) presented the growth of small (hemi)spherical gold clusters on triangular gold nanosheets. Several notable publications, on the synthesis of the platelets (followed by time-resolved SAXS), on their surface deposition for catalysis and detection applications and on the study of their vibrational modes using ultrafast X-ray diffraction (in collaboration with the group of Matias Bargheer.)

Two other talks focused on magnetic nanoparticles:

• Laura Rossi (Utrecht University) on the self-assembly of hematite cubes (paper not yet published).
• Golnaz Isapour (Fribourg University, in the group of Marco Lattuada) on color-changing materials based on responsive polymers (pNIPAM for temperature and PVP for pH).

Aside from the nice work, the last talk also references a paper on Color change in chameleons, from which I learned that structures that generate structural colors are called iridophores (great name!), in contrast with the pigment-bearing chromatophores. I have already written about structural colors on this blog.