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Nature Video
19 January 2022

Tiny feather wing beetle reveals new way to fly

How a novel flying technique copes with the weird physics of miniature flight

Adam Levy

Adam Levy

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Flying is a tricky business, but when you are less than a millimetre long, things get even tougher. At such tiny scales the physics of flight changes, and yet insects have evolved strategies to fly in miniature. In this Nature video, we zoom in on new research which is revealing the never-before-seen flight tactics of a species of feather wing beetle.Research Article: Novel flight style and light wings boost flight performance of tiny beetles

doi: https://doi.org/10.1038/d41586-022-00144-0

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EDITORIAL
19 January 2022

Biodiversity faces its make-or-break year, and research will be key

A new action plan to halt biodiversity loss needs scientific specialists to work with those who study how governments function.

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Targeted measures can help to stop extinctions, including of Père David’s deer (Elaphurus davidianus), but conserving biodiversity will also require combating climate change, cutting pollution and enhancing sustainable food systems.Credit: Staffan Widstrand/Wild Wonders of China/Nature Picture Library

Biodiversity is being lost at a rate not seen since the last mass extinction. But the United Nations decade-old plan to slow down and eventually stop the decline of species and ecosystems by 2020 has failed. Most of the plan’s 20 targets — known as the Aichi Biodiversity Targets — have not been met.The Aichi targets are part of an international agreement called the UN Convention on Biological Diversity, and member states are now finalizing replacements for them. Currently referred to as the post-2020 global biodiversity framework (GBF), the new targets are expected to be agreed this summer at the second part of the convention’s Conference of the Parties (COP15) in Kunming, China. The meeting was due to be held in May, but is likely to be delayed by a few months. Finalizing the framework will be down to government representatives working with the world’s leading biodiversity specialists. But input from social-science researchers, especially those who study how organizations and governments work, would improve its chances of success.A draft of the GBF was published last July. It aims to slow down the rate of biodiversity loss by 2030. And by 2050, biodiversity will be “valued, conserved, restored and wisely used, maintaining ecosystem services, sustaining a healthy planet and delivering benefits essential for all people”. The plan comprises 4 broad goals and 21 associated targets. The headline targets include conserving 30% of land and sea areas by 2030, and reducing government subsidies that harm biodiversity by US$500 billion per year. Overall, the goals and targets are designed to tackle each of the main contributors to biodiversity loss, which include agriculture and food systems, climate change, invasive species, pollution and unsustainable production and consumption.
Fewer than 20 extinctions a year: does the world need a single target for biodiversity?
The biodiversity convention’s science advisory body is reviewing the GBF and helping governments to decide how the targets are to be monitored. But researchers and policymakers have been writing biodiversity action plans since the 1990s, and most of these strategies have failed to make a lasting impact on two of the three key demands: that global biodiversity be conserved and that natural resources be used sustainably.Some of these failures are to do with governance, which is why it is important to involve not just researchers in the biological sciences, but also people who study organizations and how governments work. This knowledge, when allied to conservation science, will help policymakers to obtain a fuller picture of both the science gaps and the organizational challenges in implementing biodiversity plans.The GBF is a comprehensive plan. But success will require systemic change across public policy. That is both a strength and a weakness. If systemic change can be implemented, it will lead to real change. But if it cannot, there’s no plan B. This has led some researchers to argue that one target or number should be prioritized, and defined in a way that is clear to the public and to policymakers. It would be biodiversity’s equivalent of the 2 °C climate target. The researchers’ “rallying point for policy action and agreements” is to keep species extinction to well below 20 per year across all major groups (M. D. A. Rounsevell et al. Science 368, 1193–1195; 2020). Such focus does yield results. A study published in Conservation Letters found a high probability that targeted action has prevented 21–32 bird and 7–16 mammal extinctions since 1993 (F. C. Bolam et al. Conserv. Lett. 14, e12762; 2021). Extinction rates would have been around three to four times greater without conservation action, the researchers found.But not all agree that just one target should be given priority. A group of more than 50 biodiversity researchers from 23 countries point out in a policy report this week (see go.nature.com/3fv8oiv) that data on species are distributed unequally: 10, mostly high-income, countries account for 82% of records.
The United Nations must get its new biodiversity targets right
The researchers also modelled how different scenarios would affect the GBF’s 21 targets. They found that achieving the targets would require action in all of the target areas — not just a few. Focusing strongly on just one or two targets — such as expanding protected areas — will have, at best, a modest impact on achieving the UN convention’s goals and targets.The difficulty in getting governments to adopt such an integrated approach is that they (as well as non-governmental organizations and businesses) tend to tackle sustainability challenges piecemeal. Actions from last November’s climate COP in Glasgow, UK, will be implemented separately from those decided at the biodiversity COP because, in most countries, different government departments deal with climate change and biodiversity.The science advisers for the biodiversity convention will meet in Geneva, Switzerland, in March to finalize their advice. They are not advocating reform of how governments organize themselves to implement policies in sustainable development — partly (and rightly) because this is generally beyond their fields of expertise. But it’s not too late to consult those with the relevant knowledge.In the past, the UN has commissioned social scientists, for example in the UN Intellectual History Project, a series of 17 studies summarizing the experience of UN agencies spanning gender equality, diplomacy, development, trade and official statistics. However, this work, which ended in 2010, did not assess what has and hasn’t worked in science and environmental policy. Unless these perspectives are incorporated into biodiversity-research advice, any future plans risk going the way of their predecessors.

Nature 601, 298 (2022)
doi: https://doi.org/10.1038/d41586-022-00110-w

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All methods in this study were approved by the Institutional Animal Care and Use Committee at Texas Tech University (protocol 18032-12). All procedures were performed in accordance with relevant guidelines in the manuscript and the ARRIVE (Animal Research: Reporting of In Vivo Experiments) guidelines (https://arriveguidelines.org/).Experimental design for testing effects of temperature and humidity on Pd infection severity on Perimyotis subflavus
We randomly assigned bats to seven environmental chambers (Caron, Model 7000-33-1, Marietta, Ohio, USA) in a blocked experimental design, controlling temperature and humidity in each chamber (Fig. 1). In each environmental chamber, we divided bats into two cages (23 × 38 × 50 cm) constructed from mesh fabric (Part FMLF, Seattle Fabrics, Inc., Seattle, Washington, USA), PVC pipe, and plastic sheeting. We stratified random assignment to ensure even distribution of initial body mass and sex across microclimate treatments. In addition to the seven treatments with fixed temperature and humidity conditions, we had two treatments that allowed bats to freely move among temperature or humidity conditions (Fig. 1). One group of bats (n = 14) was free to move among three chambers with a common temperature (8 °C) but different humidity (water vapor pressure deficit (VPD) = 0.05 kPa, 0.10 kPa, or 0.15 kPa, corresponding to 95, 90, and 85% relative humidity (RH))36. A second group of bats (n = 14) was free to move among three chambers with a common VPD condition (0.10 kPa, medium humidity) but different temperatures (5, 8, or 11 °C) (Fig. 1). Because our research questions were focused on comparing the effect of temperature and humidity conditions on disease severity, we did not include sham-inoculated control animals in the experiment. We made this decision to reduce the total number of animals used in the experiment and to maximize replication to test the effects of temperature and humidity on disease.Figure 1Schematic of the experimental design and sample sizes with 7 environmental chambers with fixed temperature and humidity conditions and two sets of connected chambers allowing bats to behaviorally select temperature (left) or humidity conditions (bottom) for the infection trial on tri-colored bats (Perimyotis subflavus). Water loss conditions were based on water vapor pressure deficit (VPD) levels set to 0.05 kPA to produce low potential evaporative water loss (pEWL) for high humidity, 0.10 kPa for medium pEWL and humidity, or 0.15 kPA for high pEWL and low humidity. Numbers are sample sizes of bats assigned to separate cages within each chamber. Bats in the low temperature and high humidity chamber were combined into a single cage after a camera failed at the start of the experiment (top right).Full size imageWe inoculated each bat by spreading 20 µL of Pd solution (5 × 105 conidia µL−1) evenly across both wings, following established protocols8,9,32,37; treatments were conducted blind without knowledge of which bat was being assigned to what group and bats were inoculated in no particular order to reduce the confounding influence on the order of treatment. We used a Pd strain collected by Karen J. Vanderwolf at Trent University from naturally infected Myotis lucifugus. We cultured Pd on Sabouraud Dextrose Agar with chloramphenicol and gentamicin (SabDex) (Part L96359, Fisher Scientific, Houston, Texas, USA) and incubated subcultured plates at 10 °C for 60 days to allow the formation of conidia. We then harvested conidia by flooding plates with phosphate buffered saline solution containing 0.5% Tween20 (PBST). Conidia were resuspended in PBST, enumerated, and diluted to the inoculum concentration8.Microclimate treatment conditionsWe used three temperatures 5, 8, or 11 °C to represent a range of roosting temperatures of P. subflavus in natural hibernacula24,29. We set humidity in environmental chambers to achieve specific levels of water vapor pressure deficit (VPD) between the surface of the bat and the environment because relative humidity varies by temperature36. Higher VPD corresponds to drier air resulting in higher potential evaporative water loss (pEWL). We used three levels of VPD: 0.05, 0.10, or 0.15 kPa corresponding to low pEWL (high humidity), medium pEWL (medium humidity), and high pEWL (low humidity) levels (Fig. 1). We verified the ambient temperature and relative humidity in each chamber at 10-min intervals (Hobo Model U23-001, Onset Computer Corporation, Bourne, Massachussetts, USA). For bats in the connected chambers that could behaviorally select their temperature and humidity conditions, we quantified the number of days bats spent in each condition38.Animal handling and data collectionWe used 98 (42 females, 56 males) tricolored bats collected on 10 December 2018 from culverts in Mississippi and transported directly to Texas Tech University39. We took morphometric measurements (body mass ± 0.1 g, forearm length ± 0.1 mm) and used quantitative magnetic resonance (QMR; Echo-MRI-B, Echo Medical Systems, Houston, Texas, USA) to determine pre-hibernation fat at the start of the experiment39,40. As an indicator of pre-hibernation stress, we collected a fur sample from the dorsal intrascapular region to quantify fur cortisol concentration with a commercial ELISA kit, following the manufacturer’s protocol (Arbor Assays, Michigan, USA) (see Supplemental Methods). Fur is moulted once per year in the late summer period41 and therefore fur cortisol reflects the level of circulating cortisol during the period of fur growth prior to hibernation. We attached a uniquely marked, modified datalogger42 (DS1925L iButton, Maxim Integrated, San Jose, California, USA) to the back of each bat using ostomy cement to record skin temperature39. Prior to inoculation, we swabbed bats with a sterile polyester swab (Fisherbrand synthetic tipped applicators 23-400-116) five times on forearm and five times on muzzle to determine if any bats were naturally infected with Pd at time of collection. Swabs were stored in RNAlater at  − 20 °C until testing using quantitative polymerase chain reaction (qPCR) at Northern Arizona University43.During the experiment, we provided ad libitum drinking water in each cage but did not provide food. We secured a motion-activated infrared camera (Model HT5940T, Speco Technologies, New York, New York, USA) above each cage to monitor bats throughout the experiment. Because one camera failed at the start of the experiment, we combined bats in that treatment chamber into a single cage (Fig. 1) and replicated this disturbance among all chambers. We monitored bats without disturbance by reviewing video recordings daily. Three bats died of unknown cause before the end of the experiment and were removed from analyses.After 83 days of hibernation, we terminated the experiment and bats were removed from cages and processed to determine body condition using QMR39. We took respirometry measurements on a subset of animals38, and swabbed for Pd as described above. We photographed the left ventral wing using ultraviolet (UV) transillumination (368-nm wavelength and 2-s exposure) to detect and measure florescence associated with Pd infection37,44. For histology, we removed the wing section from the fifth digit and the body and rolled wing tissue around dental wax dowels and 10% neutral buffered formalin. We collected a 90–110 µL blood sample in lithium-heparin-treated capillary tubes for immediate analysis of blood chemistry with a handheld analyzer (i-STAT1 Vet Scan, Abaxis, Union City, California, USA). Using an EC8+ cartridge, we measured sodium, potassium, chloride, anion gap, glucose, BUN (urea nitrogen), hematocrit, hemoglobin, pH, pCO2, TCO2, HCO3, and base excess (Table S1). We quantified arousals from torpor as reported by McGuire et al.39. All bats were handled and euthanized under Animal Care and Use Committee permit 18032-12 at Texas Tech University.Infection and disease metricsWe used several metrics to determine pathogen and disease presence and severity37: presence and amount of the pathogen, Pd, on a bat were determined by qPCR43, and presence of the disease, WNS, was determined via detection of orange-yellow florescence under UV light characteristic of Pd infection44 and histological presence of characteristic lesions and pustules with fungal hyphae45,46. Three types of cutaneous infection were described histologically, including characteristic cupping erosions with fungal hyphae, neutrophilic pustules with fungal hyphae, and fungal hyphae in the stratum corneum with dermal necrosis. Any bats with any of these three conditions noted were scored as WNS positive by histology. Presence and quantity of DNA of Pd was tested by qPCR at Northern Arizona University. All samples were run in duplicate and considered positive if at least one run was positive below a cycle threshold (Ct) of 40 and quantified using a quantification curve from serial dilutions (nanograms of Pd using the equation load = 10((22.049-Ct value)/3.34789), r2 = 0.986)47. Load values were averaged across multiple runs and then converted to attograms by multiplying loads in nanograms by 109.Statistical analysesWe used three different response variables (Pd prevalence, Pd loads, and WNS prevalence by histology) to determine whether infection status varied by microclimate treatment conditions. Low sample sizes of positive infection status by UV detection (n = 4) precluded use in statistical analyses (Table 1). We used generalized linear models with binomial distribution for analyses of Pd prevalence and WNS prevalence and a linear mixed effects model with Gaussian errors for Pd loads. Although the experiment was designed with replication at the cage level to account for cage effects, we were unable to include cage as a random effect because of the low numbers of bats that had signs of Pd or WNS infection. We analyzed whether infection status (i.e., Pd prevalence, Pd load, or WNS prevalence) varied by sex and cortisol separately from an a priori candidate model set (Table 2) to cope efficiently with small sample sizes. We first asked whether infection response varied by sex to determine if bats could be pooled in subsequent analyses. We analyzed separately whether infection response varied by pre-hibernation cortisol at the start of the experiment on the subset of animals for which we had cortisol measurements (n = 83). We then used an information-theoretic approach comparing a candidate set of models with Akaike Information Criterion (AIC)48 using initial fat mass as an individual covariate and temperature and humidity treatment conditions as categorical treatment groups to assess the effect of microclimate on infection response (Table 2). Bats behaviorally selecting their temperature and humidity conditions were assigned to a temperature or humidity treatment level if a bat spent  > 89% of captive days at that condition or was otherwise placed in an ‘inconstant condition’ treatment group. For WNS prevalence, we used the bias reduction method implemented in package brglm49 to deal with complete separation present in the data (in some treatments all bats were scored as negative for WNS) (Table 1; Fig. 2).Table 1 Signs of Pd infection or WNS disease for tri-colored bats (Perimyotis subflavus) exposed to different temperature and humidity regimes.Full size tableTable 2 Model selection results for model comparisons of humidity and temperature and pre-hibernation fat mass on Pd prevalence, Pd load, and WNS prevalence.Full size tableFigure 2Signs of Pseudogymnoascus destructans (Pd) infection or white-nose syndrome (WNS) disease for tri-colored bats (Perimyotis subflavus) exposed to different temperature and humidity regimes. (A) Fraction of bats with Pd detected by qPCR; (B) Fraction of bats with signs of WNS disease by histology, and (C) Mean quantity of Pd on bats at the end of the experiment. There was no statistical support for differences between temperature or humidity treatments for any response metrics. Points are estimated means and vertical lines show binomial standard error for prevalence and standard errors for Pd load.Full size imageBecause this was the first captive hibernation experiment with P. subflavus, we investigated the effects of temperature and humidity on the hibernation physiology of the species38,39 and how physiological markers (e.g., blood chemistry) may be associated with disease. To determine if physiological indicators were related to infection status at the end of the experiment, we compared total number of torpor arousal bouts during the experiment and 13 different blood chemistry metrics from blood samples taken at the end of the experiment and used t-test comparisons (at α = 0.05) for each metric between Pd/WNS positive and negative bats. We designated bats as Pd/WNS positive if a bat tested positive for either Pd or WNS by qPCR, UV, or histology. We used Program R version 3.6.2 to conduct all analyses.Experimental design for testing effects of temperature and humidity on Pd growth on substratesWe used five environmental chambers (CARON, Model 7000-33-1, Marietta, Ohio, USA) to test for the effects of temperature and humidity on fungal growth on natural and artificial substrates (Fig. S1). Our experimental design comprised a reduced temperature series and humidity gradient than what we used for the experiment on bats. In the humidity gradient, temperature was held constant at 8 °C, with 85%, 90%, and 95% RH representing our low, medium, and high humidity treatments, respectively. In the temperature series, vapor pressure deficit (VPD) was held constant across the low (5 °C), medium (8 °C), and high (11 °C) temperatures (VPD = nominally 0.01 kPa, range (0.105–0.107). The chamber set to 8 °C and 90% humidity (VPD = 0.107 kPa) was common to both series.Media plate inoculation and fungal growth measurementWe constructed modified plate lids to prevent contamination while allowing humidity to equilibrate across the plate lid. We drilled 14 equidistant holes (5.5 mm diameter) into each plate lid and hot glued a piece of circular filter paper to the top of the lid. Lids were then disinfected thoroughly with a hydrogen peroxide wipe before being placed in a disinfected, sealed storage container.We prepared Pd inoculum as described above for the infection trial on bats. We inoculated 30 SabDex plates with 100 µL of inoculum at a concentration of 20 conidia µL−1 by serial dilution with a starting concentration of 2.0 × 104 conidia µL−1 diluted four times by a factor of 10. We used sterile, individually wrapped 1-µL plastic inoculation loops to spread the inoculum evenly across the surface of the plates, added the modified plate lids, and immediately transferred plates into environmental chambers. We included six replicate plates in each of the five microclimate conditions.We took weekly digital photographs (Nikon, Model 26524, Tokyo, Japan) of each plate for the 5-week duration of the experiment (Fig. 3A). Our camera was mounted on a tripod to ensure consistent placement of plates relative to the camera. Each photo included a ruler, which was used to calibrate measurements made in ImageJ (Version 2.0.0-rc-69/1.52p, National Institutes of Health, Bethesda, Maryland, USA). One observer made all measurements for consistency. We used the freehand selection tool to trace the boundary of each fungal colony using a drawing tablet (Wacom, Model CTL-490, Kazo, Saitama, Japan). From these selections, we obtained the total surface area growth as the sum of all area selection (in cm2).Figure 3Examples demonstrate the process of measuring and estimating fungal growth of Pseudogymnoascus destructans (Pd) on media plates in temperature and humidity treatment conditions. (A) Examples of fungal growth on media plates measured at days 7, 14, 21, 28, and 34 from two of the treatment conditions (11 °C, 92% RH and 5 °C, 88% RH). (B) Examples of estimating maximum growth rate and latency variables from fungal growth measurements in panel A. We fit a sigmoidal curve to describe fungal growth (thick solid black line) to estimate the inflection point of the curve (vertical solid line). We calculated the slope (solid red line) at the inflection point of the curve to estimate maximum growth rate, and the days until total growth area reached 2.5 cm2 (dashed red lines) as an estimate of latency.Full size imageWe modelled the growth of Pd on each plate as a sigmoidal curve (Fig. 3B), which we fit using the SSlogis and nls functions in Program R v. 3.6.350. The model fitting function provides an estimate of the inflection point of the curve, and we calculated the slope at the inflection point to estimate the maximum growth rate. We also estimated the latency to rapid fungal growth on the plates by determining the date at which the total area of fungus on the plate reached 2.5 cm2 as an arbitrary threshold.We also quantified growth of individual colonies. To avoid biasing growth rate estimates, we excluded colonies that intercepted another colony by choosing independent colonies at the final time point and tracking them backwards through time. If there were fewer than 10 independent colonies at the final time point, we added additional unimpeded colonies with each earlier time point until the total number of colonies reached 10. We modelled growth of individual colonies following the same procedure as for total area of growth on the plate, with an arbitrary threshold of 0.05 cm2 for latency calculations. We used linear mixed models to test for the effects of temperature and humidity on maximum growth rate or latency, including plate as a random factor to account for measuring multiple colonies per plate.Rock inoculation and fungal growth measurementTo evaluate fungal growth and persistence on a natural substrate, we inoculated pieces of sandstone flagstone. We etched a 4 × 6 sampling grid, composed of 5 × 5 cm squares, onto the surface of each sandstone rock (Texas Rock and Flagstone, Lubbock, Texas, USA), where each square served as a sampling unit (Fig. S2). Each row represented a time series for a single replicate, while each column was composed of replicates for the respective time point. Rocks were then autoclaved at 121 °C for 40 min and stored individually in a disinfected, sealed container until inoculation. At the time of inoculation, we evenly spread 200 µL of inoculum (2.5 × 104 conidia µL−1) across each sampling square and immediately transferred the rock to an environmental chamber.We measured fungal growth at days 0, 14, 28, and 56. We used a sterile cotton swab to collect fungal DNA from each sampling square. Swabs were moistened with RNAlater and rolled horizontally, vertically, and diagonally across the surface of the sampling square to ensure contact with the total surface area. One researcher collected all swabs to maximize consistency among swabs collected throughout the experiment. Swabs were placed in RNAlater and stored at − 20 °C until shipped to Northern Arizona University for qPCR analysis43. We quantified fungal loads for each swab sample from qPCR using the quantification curve provided above and normalized fungal loads to the value at day zero for each rock respectively. We then used linear models to test for effects of temperature and humidity on changes in fungal load (log transformed) over time.To evaluate viability of Pd, we swabbed the entire inoculated surface of each rock at the end of the experiment and vortexed the swabs in RNAlater for one minute to release fungal DNA from the swab. We then applied 100 µL of RNAlater fungal solution from each rock to a respective SabDex media plate, using a sterile inoculation loop. After 2 weeks of incubation at 11 °C and 92% RH, we visually assessed plates for presence of fungal growth to determine viability of Pd collected from rocks at the end of the growth experiment. More

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How many parameters best describe data in muon spectroscopy?Here we find that the Bayes factor demands the inclusion of more physically-meaningful parameters than the BIC or significance tests. Figure 1a presents some data that might reasonably be fitted with as few as three or as many as 22 physically-meaningful parameters. We find that the Bayes factor encourages the inclusion of all these parameters until the onset of over-fitting. Even though many of them have fitted values that fail significance tests (i.e. are consistent with zero), their omission distorts the fitting results severely.Figure 1Full size imageFigure 1a shows an anti-level-crossing spectrum observed in photo-excited muon-spin spectroscopy26 from an organic molecule27. The data are presented in Fig. 2a of Ref.27 and are given in the SI. These spectra are expected to be Lorentzian peaks. Theory permits optical excitation to affect the peak position, the width and the strength (photosensitivity). In the field region over which the measurements are carried out, there is a background from detection of positrons, which has been subtracted from the data presented27. Wang et al.27 did not attempt to fit the data rigorously; they did report a model-independent integration of the data, which demonstrated a change in area and position.The model that we fit hypothesises one or more Lorentzian peaks, with optional photosensitivity on each fitting parameter and with optional linear backgrounds y = a + bx underlying the peaks, described by the full equation given in the SI, equation (S3). To do a single LS fit to all the data, we extend the data to three dimensions, (x gauss, y asymmetry, z) where z = 0 for data in the dark and z = 1 for photoexcited data. Including all the data in a single LS fit in this way, rather than fitting the dark and photoexcited data separately, simplifies both setting up the fit and doing the subsequent analysis.Figure 1b shows the evolution of the SBIC and the lnBF as the number of fitting parameters in the model is increased. Starting with a single Lorentzian peak, three parameters are required, peak position P, width W and intensity A. Three photosensitivity parameters ΔLP, ΔLW and ΔLA are then introduced successively to the fit, (open and small data points for n = 3–6). The SBIC decreases and the lnMLI scarcely increases. It is only with the inclusion of one background term (n = 7) that any figure of merit shows any substantial increase. There is no evidence here for photosensitivity. The weak peak around 7050 G does not seem worth including in a fit, as it is evidenced by only two or three data points and is scarcely outside the error bars. However, a good fit with two peaks (P1 ~ 7210 G, P2 ~ 7150 G, the subscripts 1 and 2 in accordance with the site labelling of Fig. 2a of Ref.27) can be obtained with just five parameters (P1, P2, A1, A2, W). This gives substantial increases in the SBIC and lnMLI, further increased when W1 and W2 are distinguished and then when the single background term and the three photosensitivity parameters ΔLP2, ΔLW2 and ΔLA2 are successively included (solid or large data points for n = 5–10 in Fig. 1b). The SBIC reaches its maximum here, at n = 10, and then decreases substantially when the other three photosensitivity parameters and the other three background terms are included. These additional parameters fail significance tests as well as decreasing the SBIC (Fig. 1b). Conventionally, the n = 10 fit would be accepted as best. The outcome would be reported as two peaks, with significant photo-sensitivities ΔLP2, ΔLW2 and ΔLA2 for all three of the 7150 G peak parameters, but no photosensitivity for the 7210 G peak (Table 1).Table 1 Photosensitivity results of fitting the data of Fig. 1a with 10, 16 and 19 parameters. Parameter units as implied by Fig. 1a.Full size tableThe Bayes factor gives a very different outcome. From 10 to 16 parameters, the Bayes factor between any two of these seven models is close to unity (Fig. 1b). That is, they have approximately equal probability. The Bayes factor shows that what the conventional n = 10 analysis would report is false. Specifically, it is not the case that ΔLP2, reported as − 14 ± 4 G, has a roughly ({raise0.5exhbox{$scriptstyle 2$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle 3$}}) probability of lying between − 10 and − 18 G. That is not consistent with the roughly equal probability that it lies in the n = 16 range (− 24 ± 8 G). Table 1 shows that at n = 16, ΔLP2 is the only photosensitivity parameter to pass significance tests. ΔLA2, which had the highest significance level at n = 10, is now the parameter most consistent with zero. The other four are suggestively (about 1({raise0.5exhbox{$scriptstyle 1$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle 2$}})σ) different from zero.Since the Bayes factor has already radically changed the outcome by encouraging more physically-meaningful parameters, it is appropriate to try the 7050 G peak parameters in the fit. With only 28 data-points, we should be alert to over-fitting. We can include P3 and A3 (n = 18), and ΔLP3 (n = 19), but W3 and ΔLA3 do cause overfitting. Figure 1b shows substantial increases of both the SBIC and the lnMLI for n = 18 to n = 20, where the twentieth parameter is in fact ΔLA3. The symptom of over-fitting that we observe here is an increase in the logarithm of the Occam Factor (lnMLI − lnL), the values of which decrease, − 26.9, − 33.5, − 34.8, and then increase, − 33.4, for n = 16, 18, 19 and 20 respectively. Just as lnL must increase with every additional parameter, so should the Occam factor decrease, as the prior parameter volume should increase more with a new parameter than the posterior parameter volume. So we stop at n = 19. The outcome, Table 1, is that the uncertainties on the n = 16 parameters have decreased markedly. This is due to the better fit, with a substantial increase in lnL corresponding to reduced residuals on all the data. The 7210 G peak 2 now has photosensitivities on all its parameters, significant to at least the 2σ or p value ~ 0.05 level. And the photosensitivities ΔLW2 and ΔLA2, both so significant at n = 10, and already dwindling in significance at n = 16, are both now taking values quite consistent with zero. In the light of Table 1, we see that stopping the fit at n = 10 results in completely incorrect results—misleading fitted values, with certainly false uncertainties.Discriminating between models for the pressure dependence of the GaAs bandgapThe main purpose of this example is to show how the Bayes factor can be used to decide between two models which have equal goodness of fit to the data (equal values of lnL and BIC, as well as p values, etc.). This illustrates the distinction it makes between physically-meaningful and physically meaningless parameters. This example also shows how ML fitting can be used together with the Bayes factor to obtain better results. For details, see SI §7.Figure 2 shows two datasets for the pressure dependence of the bandgap of GaAs (data given in the SI). The original authors published quadratic fits, ({E}_{g}(P)={E}_{0}+bP+c{P}^{2}), with b = 10.8 ± 0.3 meV kbar−1 (Goñi et al.28) and 11.6 ± 0.2 meV kbar−1 (Perlin et al.29). Other reported experimental and calculated values for b ranged from 10.02 to 12.3 meV kbar−130. These discrepancies of about ± 10% were attributed to experimental errors in high-pressure experimentation. However, from a comparison of six such datasets, Frogley et al.30 were able to show that the discrepancies arose from fitting the data with the quadratic formula. The different datasets were reconciled by using the Murnaghan equation of state and supposing the band-gap to vary linearly with the density (see SI, §7, equations (S4) and (S5)30. The curvature c of the quadratic is constant, while the curvature of the density, due to the pressure dependence Bʹ of the bulk modulus B0, decreases with pressure—and the six datasets were recorded over very different pressure ranges, as in Fig. 2. So the fitted values of c, c0, were very different, and the correlation between b and c resulted in the variations in b0.Here, using the Bayes factor, we obtain the same result from a single dataset, that of Goñi et al.28 The two fits are shown in Fig. 2. They are equally good, with values of lnL and SBIC the same to 0.01. The key curvature parameters, c and ({text{B}}^{prime }), are both returned as non-zero by 13.5σ (SI, §7, Table S1), consequently both with p-values less than 10−18. However, c is a physically-meaningless parameter. The tightest constraint we have for setting its range is the values previously reported, ranging from 0 to 60 μeV kbar−2, so we use Δc = 100 μeV kbar−2. In contrast, ({text{B}}^{prime }) is known for GaAs to be 4.4931. For many other materials and from theory the range 4–5 is expected, so we use (Delta {text{B}}^{prime } = 1). The other ranges are same for both models (see SI §7). This difference gives a lnBF of 3.8 in favour of the Murnaghan model against the quadratic, which is strong evidence for it. Moreover, the value of ({text{B}}^{prime }) returned is 4.47 ± 0.33, in excellent agreement with the literature value. Had it been far out of range, the model would have to be rejected. The quadratic model is under no such constraint; indeed, a poor fit might be handled by adding cubic and higher terms ad lib. This justifies adding about 5 to lnBF (see “Background in fitting a carbon nanotube Raman spectrum” section), giving a decisive preference to the Murnaghan model, and the value of b it returns, 11.6 ± 0.3. Note the good agreement with the value from Perlin et al.29 If additionally we fix ({mathrm{B}}^{prime}) at its literature value of 4.4931, lnBF is scarcely improved, because the Occam factor against this parameter is small, but the uncertainty on the pressure coefficient, Ξ/B0, is much improved.When we fit the Perlin data, the Murnaghan fit returns ({text{B}}^{prime }) = 6.6 ± 2.4. This is outside range, and indicates that this data cannot give a reliable value—attempting it is over-fitting. However, it is good to fit this data together with the Goñi data. The Perlin data, very precise but at low pressures only, complement the Goñi data with their lower precision but large pressure range. We notice also that the Perlin data has a proportion of outlier data points. Weighted or rescaled LS fitting can handle the different precisions, but it cannot handle the outliers satisfactorily. Maximum Likelihood fitting handles both issues. We construct lnL using different pdfs P(r) for the two datasets, and with a double-Gaussian pdf for the Perlin data (see equation (S6) in the SI §7). Fixing ({text{B}}^{prime }) at 4.49, fitting with the same Ξ/B0 returns 11.42 ± 0.04 meV kbar−1. Separate Ξ/B0 parameters for the two datasets give an increase of lnL of 4.6, with values 11.28 ± 0.06 and 11.60 ± 0.04 meV kbar−1—a difference in b of 0.32 ± 0.07 meV kbar−1, which is significant at 4½σ. This difference could be due to systematic error, e.g. in pressure calibration. Or it could be real. Goñi et al.28 used absorption spectroscopy to measure the band-gap; Perlin et al.29 used photoluminescence. The increase of the electron effective mass with pressure might give rise to the difference. In any case, it is clear that high-pressure experimentation is much more accurate than previously thought, and that ML fitting exploits the information in the data much better than LS fitting.Figure 2GaAs band-gap. Data for Eg(P) in GaAs from Goñi et al.28 (
) and from Perlin et al.29 (
) are shown after subtraction of the straight line E0 + 8.5P to make the curvature more visible. The Perlin data is expanded × 10 on both axes for clarity. Two least-squares fits to the Goñi data are shown, polynomial (dashed red line) and Murnaghan (solid blue line). (Figure prepared using Mathematica 12.0, www.wolfram.com/mathematica/).Full size imageBackground in fitting a carbon nanotube Raman spectrumThis example demonstrates how the Bayes Factor provides a quantitative answer to the problem, whether we should accept a lower quality of fit to the data if the parameter set is intuitively preferable. It also provides a simple example of a case where the MLI calculated by Eq. (1) is in error and can readily be corrected (see SI §8 Fig. S3).The dataset is a Raman spectrum of the radial breathing modes of a sample of carbon nanotubes under pressure32. The whole spectrum at several pressures is shown with fits in Fig. 1 of Ref.32. The traditional fitting procedure used there was to include Lorentzian peaks for the clear peaks in the spectra, and then to add broad peaks as required to get a good fit, but without quantitative figures of merit and without any attempt to explain the origin of the broad peaks, and therefore with no constraints on their position, widths or intensities. The key issue in the fitting was to get the intensities of the peaks as accurately as possible, to help understand their evolution with pressure. Here, we take a part of the spectrum recorded at 0.23 GPa (the data is given in the SI.) and we monitor the quality of fit and the Bayes factor while parameters are added in four models. This part of the spectrum has seven sharp pseudo-Voigt peaks (Fig. 3a; the two strong peaks are clearly doublets). With seven peak positions Pi, peak widths Wi and peak intensities Ai, and a factor describing the Gaussian content in the pseudo-Voigt peak shape, there are already 22 parameters (for details, see SI §8). This gives a visibly very poor fit, with lnL = − 440, SBIC = − 510 and lnMLI = − 546. The ranges chosen for these parameters for calculating the MLI (see SI §8) are not important because they are used in all the subsequent models, and so they cancel out in the Bayes factors between the models.Figure 3Carbon nanotube Raman spectrum. In (a), the carbon nanotube Raman spectrum is plotted (black datapoints) with a fit (cyan solid line) using the Fourier model. The residuals for four good fits are shown, × 10 and displaced successively downwards (Fourier, Polynomial, Peaks and Tails; all at lnL about − 60, see text). The backgrounds are shown, × 8 (long dashed, chain-dotted, short dashed and solid, respectively. In (b), the evolution of the MLIs is shown against the number of parameters for these four models. (Figure prepared using Mathematica 12.0, www.wolfram.com/mathematica/).Full size imageTo improve the fit, in the Fourier model we add a Fourier background (y=sum {c}_{i}mathrm{cos}ix+{s}_{i}mathrm{sin}ix) (i = 0,..) and in the Polynomial model, we add (y=sum {a}_{i}{x}^{i}) (i = 0,..) for the background. In both, the variable x is centred (x = 0) at the centre of the fitted spectrum and scaled to be ± π or ± 1 at the ends. In the Peaks model we add extra broad peaks as background, invoking extra parameter triplets (Pi, Wi, Ai). These three models all gave good fits; at the stage shown in Fig. 3a they gave lnL values of − 65, − 54 and − 51 and BIC values of − 156, − 153 and − 148 respectively. Thus there is not much to choose between the three models, but it is noteworthy that they give quite different values for the intensities of the weaker peaks, with the peak at 265 cm−1 at 20.5 ± 1.1, 25.5 ± 1.3 and 27 ± 1.7 respectively (this is related to the curvature of the background function under the peak). So it is important to choose wisely.A fourth model was motivated by the observation that the three backgrounds look as if they are related to the sharp peaks, rather like heavily broadened replicas (see Fig. 3a). Accordingly, in the fourth model, we use no background apart from the zeroth term c0 or a0 to account for dark current). Instead, the peak shape is modified, giving it stronger, fatter tails than the pseudo-Voigt peaks (Tails model). This was done by adding to the Lorentzian peak function a smooth function approximating to exponential tails on both sides of the peak position (for details, see SI §8) with widths and amplitudes as fitting parameters. What is added may be considered as background and is shown in Fig. 3a. This model, at the stage of Fig. 3a, returned lnL = − 62, BIC = − 146, and yet another, much smaller value of 15.5 ± 1.0 for the intensity of the 265 cm−1 peak.The Tails model is intuitively preferable to the other three because it does not span the data space—e.g. if there was really were broad peaks at the positions identified by the Peaks model, or elsewhere, the Tails model could not fit them well. That it does fit the data is intuitively strong evidence for its correctness. The Bayes factor confirms this intuition quantitatively. At the stage of Fig. 3a, the lnMLI values are − 251, − 237 and − 223 for the Fourier, Poly and Peaks models, and − 211 for the Tails model. This gives a lnBF value of 12 for the Tails model over the Peaks model—decisive—and still larger lnBF values for these models over the Fourier and Poly models.All models can be taken further, with more fitting parameters. More Fourier or polynomial terms or more peaks can be added, and for the Tails model more parameters distinguishing the tails attached to each of the seven Lorentizian peaks. In this way, the three background models can improve to a lnL ~ − 20; the Tails model does not improve above lnL ~ − 50. However, as seen in Fig. 3b, the MLIs get worse with too many parameters, except when over-fitting occurs, as seen for the Poly model at 35 parameters. The Tails model retains its positive lnBF  > 10 over the other models.The other models can have an indefinite number of additional parameters—more coefficients or more peaks, to fit any data set. It is in this sense that they span the data space. The actual number used is therefore itself a fitting parameter, with an uncertainty perhaps of the order of ± 1, and a range from 0 to perhaps a quarter or a half of the number of data points m. We may therefore penalise their lnMLIs by ~ ln 4 m−1 or about − 5 for a few hundred data points. This takes Tails to a lnBF  > 15 over the other models—overwhelmingly decisive. This quantifies the intuition that a model that is not guaranteed to fit the data, but which does, is preferable to a model that certainly can fit the data because it spans the data space. It quantifies the question, how much worse a quality of fit should we accept for a model that is intuitively more satisfying. Here we accept a loss of − 30 on lnL for a greater gain of + 45 in the Occam factor. It quantifies the argument that the Tails model is the most worthy of further investigation because the fat tails probably have a physical interpretation worth seeking. In this context, it is interesting that in Fig. 3a fat tails have been added only to the 250, 265 and 299 cm−1 peaks; adding fat tails to the others did not improve the fit; however, a full analysis and interpretation is outside the scope of this paper. In the Peaks model it is not probable (though possible) that the extra peaks would have physical meaning. In the other two models it is certainly not the case that their Fourier or polynomial coefficients will have physical meaning. More