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We conduct a thorough analysis of the relationship between the out-of-sample performance and the Bayesian evidence (marginal likelihood) of Bayesian neural networks (BNNs), as well as looking at the performance of ensembles of BNNs, both using the Boston housing dataset. Using the state-of-the-art in nested sampling, we numerically sample the full (non-Gaussian and multimodal) network posterior and obtain numerical estimates of the Bayesian evidence, considering network models with up to 156 trainable parameters. The networks have between zero and four hidden layers, either $\tanh$ or $ReLU$ activation functions, and with and without hierarchical priors. The ensembles of BNNs are obtained by determining the posterior distribution over networks, from the posterior samples of individual BNNs re-weighted by the associated Bayesian evidence values. There is good correlation between out-of-sample performance and evidence, as well as a remarkable symmetry between the evidence versus model size and out-of-sample performance versus model size planes. Networks with $ReLU$ activation functions have consistently higher evidences than those with $\tanh$ functions, and this is reflected in their out-of-sample performance. Ensembling over architectures acts to further improve performance relative to the individual BNNs., https://github.com/PolyChord/PolyChordLite; https://github.com/SuperKam91/bnn

Metropolis Hastings nested sampling evolves a Markov chain, accepting new points along the chain according to a version of the Metropolis Hastings acceptance ratio, which has been modified to satisfy the nested sampling likelihood constraint. The geometric nested sampling algorithm I present here is based on the Metropolis Hastings method, but treats parameters as though they represent points on certain geometric objects, namely circles, tori and spheres. For parameters which represent points on a circle or torus, the trial distribution is "wrapped" around the domain of the posterior distribution such that samples cannot be rejected automatically when evaluating the Metropolis ratio due to being outside the sampling domain. Furthermore, this enhances the mobility of the sampler. For parameters which represent coordinates on the surface of a sphere, the algorithm transforms the parameters into a Cartesian coordinate system before sampling which again makes sure no samples are automatically rejected, and provides a physically intuitive way of the sampling the parameter space., Peer reviewed and published in JOSS. arXiv admin note: substantial text overlap with arXiv:1905.09110

Metropolis nested sampling evolves a Markov chain from a current livepoint and accepts new points along the chain according to a version of the Metropolis acceptance ratio modified to satisfy the likelihood constraint, characteristic of nested sampling algorithms. The geometric nested sampling algorithm we present here is a based on the Metropolis method, but treats parameters as though they represent points on certain geometric objects, namely circles, tori and spheres. For parameters which represent points on a circle or torus, the trial distribution is `wrapped' around the domain of the posterior distribution such that samples cannot be rejected automatically when evaluating the Metropolis ratio due to being outside the sampling domain. Furthermore, this enhances the mobility of the sampler. For parameters which represent coordinates on the surface of a sphere, the algorithm transforms the parameters into a Cartesian coordinate system before sampling which again makes sure no samples are automatically rejected, and provides a physically intutive way of the sampling the parameter space. We apply the geometric nested sampler to two types of toy model which include circular, toroidal and spherical parameters. We find that the geometric nested sampler generally outperforms \textsc{MultiNest} in both cases. \\ %We also apply the algorithm to a gravitational wave detection model which includes circular and spherical parameters, and find that the geometric nested sampler and \textsc{MultiNest} appear to perform equally well as one another. Our implementation of the algorithm can be found at \url{https://github.com/SuperKam91/nested_sampling}., Comment: 13 pages, 11 figures, 28 equations

We present a comparison of mass estimates for $54$ galaxy cluster candidates from the second Planck catalogue (PSZ2) of Sunyaev-Zel'dovich sources. We compare the mass values obtained with data taken from the Arcminute Microkelvin Imager (AMI) radio interferometer system and from the Planck satellite. The former of these uses a Bayesian analysis pipeline that parameterises a cluster in terms of its physical quantities, and models the dark matter & baryonic components of a cluster using NFW and GNFW profiles respectively. Our mass estimates derived from Planck data are obtained from the results of the Bayesian detection algorithm PowellSnakes (PwS), are based on the methodology detailed in the PSZ2 paper, and produce two sets of mass estimates; one estimate is calculated directly from the angular radius $��$ - integrated Comptonisation parameter $Y$ posterior distributions, and the other uses a `slicing function' to provide information on $��$ based on X-ray measurements and previous Planck mission samples. We find that for $37$ of the clusters, the AMI mass estimates are lower than both values obtained from Planck data. However the AMI and slicing function estimates are within one combined standard deviation of each other for $31$ clusters. We also generate cluster simulations based on the slicing-function mass estimates, and analyse them in the same way as we did the real AMI data. We find that inclusion in the simulations of radio-source confusion & CMB noise and measurable radio-sources causes AMI mass estimates to be systematically low., 16 pages, 15 figures, 5 tables, 18 equations, MNRAS

We derive a model for Sunyaev--Zel'dovich data from a galaxy cluster which uses an Einasto profile to model the cluster's dark matter component. This model is similar to the physical models for clusters previously used by the Arcminute Microkelvin Imager (AMI) consortium, which model the dark matter using a Navarro-Frenk-White (NFW) profile, but the Einasto profile provides an extra degree of freedom. We thus present a comparison between two physical models which differ only in the way they model dark matter: one which uses an NFW profile (PM I) and one that uses an Einasto profile (PM II). We illustrate the differences between the models by plotting physical properties of clusters as a function of cluster radius. We generate AMI simulations of clusters which are \textit{created} and \textit{analysed} with both models. From this we find that for 14 of the 16 simulations, the Bayesian evidence gives no preference to either of the models according to the Jeffreys scale, and for the other two simulations, weak preference in favour of the correct model. However, for the mass estimates obtained from the analyses, the values were within $1\sigma$ of the input values for 14 out of 16 of the clusters when using the correct model, but only in 6 out of 16 cases when the incorrect model was used to analyse the data. Finally we apply the models to real data from cluster A611 obtained with AMI, and find the mass estimates to be consistent with one another except in the case of when PM II is applied using an extreme value for the Einasto shape parameter., Comment: 14 pages, 9 figures, 3 tables, 28 equations, MNRAS 2019

We develop a Bayesian method of analysing Sunyaev-Zel'dovich measurements of galaxy clusters obtained from the Arcminute Microkelvin Imager (AMI) radio interferometer system and from the Planck satellite, using a joint likelihood function for the data from both instruments. Our method is applicable to any combination of Planck data with interferometric data from one or more arrays. We apply the analysis to simulated clusters and find that when the cluster pressure profile is known a-priori, the joint dataset provides precise and accurate constraints on the cluster parameters, removing the need for external information to reduce the parameter degeneracy. When the pressure profile deviates from that assumed for the fit, the constraints become biased. Allowing the pressure profile shape parameters to vary in the analysis allows an unbiased recovery of the integrated cluster signal and produces constraints on some shape parameters, depending on the angular size of the cluster. When applied to real data from Planck-detected cluster PSZ2 G063.80+11.42, our method resolves the discrepancy between the AMI and Planck $Y$-estimates and usefully constrains the gas pressure profile shape parameters at intermediate and large radii., 13 pages, 10 figures. Submitted to MNRAS. Post-referee version

The Arcminute Microkelvin Imager (AMI) carried out a blind survey for galaxy clusters via their Sunyaev-Zel'dovich effect decrements between 2008 and 2011. The first detection, known as AMI-CL J0300+2613, has been reobserved with AMI equipped with a new digital correlator with high dynamic range. The combination of the new AMI data and more recent high-resolution sub-mm and infra-red maps now shows the feature in fact to be a ring of positive dust-correlated Galactic emission, which is likely to be anomalous microwave emission (AME). If so, this is the first completely blind detection of AME at arcminute scales., 11 pages, 13 figures, 4 tables. Submitted to MNRAS

We present a comparison between three cluster models applied to data obtained by the Arcminute Microkelvin Imager radio interferometer system. The physical model (PM) parameterises a cluster in terms of its physical quantities to model the dark matter and baryonic components of the cluster using NFW and GNFW profiles respectively. The observational models (OM I and OM II) model only the gas content of the cluster. The two OMs vary only in the priors they use in Bayesian inference: OM I has a joint prior on angular radius $\theta$ and integrated Comptonisation $Y$, derived from simulations, while OM II uses separable priors on $\theta$ and $Y$ which are based on calculations of the physical model. For the comparison we consider a sample of $54$ clusters which are a subsample of the second Planck catalogue of Sunyaev-Zel'dovich sources. We first compare the $Y$ estimates of the three models, and find that the PM generally yields lower estimates relative to the OMs. We then compute the Earth Mover's Distance between the $\theta$ - $Y$ posterior distributions obtained from each model for each cluster, and find that the two models which are most discrepant are PM and OM I. Finally, we compare the Bayesian evidence values obtained from each model for each cluster. OM I generally provides the best fit to the data but not at a statistically significant level, according to the Jeffreys scale. The highest evidence ratio obtained is actually in favour of the PM over OM I., Comment: 12 pages, 6 figures, 9 tables, 12 equations, MNRAS

We derive a model for Sunyaev–Zel’dovich data from a galaxy cluster that uses an Einasto profile to model the cluster’s dark matter component. This model is similar to the physical models for clusters previously used by the Arcminute Microkelvin Imager (AMI) consortium, which model the dark matter using a Navarro–Frenk–White (NFW) profile, but the Einasto profile provides an extra degree of freedom. We thus present a comparison between two physical models which differ only in the way they model dark matter: one which uses an NFW profile (PM I) and one that uses an Einasto profile (PM II). We illustrate the differences between the models by plotting physical properties of clusters as a function of cluster radius. We generate AMI simulations of clusters that are created and analysed with both models. From this we find that for 14 of the 16 simulations, the Bayesian evidence gives no preference to either of the models according to the Jeffreys scale, and for the other two simulations, weak preference in favour of the correct model. However, for the mass estimates obtained from the analyses, the values were within 1σ of the input values for 14 out of 16 of the clusters when using the correct model, but only in 6 out of 16 cases when the incorrect model was used to analyse the data. Finally, we apply the models to real data from cluster A611 obtained with AMI, and find the mass estimates to be consistent with one another except in the case of when PM II is applied using an extreme value for the Einasto shape parameter.

We present a comparison of mass estimates for 54 galaxy cluster candidates from the second Planck catalogue (PSZ2) of Sunyaev-Zel'dovich sources. We compare the mass values obtained with data taken from the Arcminute Microkelvin Imager (AMI) radio interferometer system and from the Planck satellite. The former of these uses a Bayesian analysis pipeline that parametrizes a cluster in terms of its physical quantities, and models the dark matter and baryonic components of a cluster using Navarro-Frenk-White (NFW) and generalized-NFW profiles, respectively. Our mass estimates derived from Planck data are obtained from the results of the Bayesian detection algorithm PowellSnakes, are based on the methodology detailed in the PSZ2 paper, and produce two sets of mass estimates; one estimate is calculated directly from the angular radius θ-integrated Comptonization parameter Y posterior distributions, and the other uses a 'slicing function' to provide information on θ based on X-ray measurements and previous Planck mission samples. We find that for 37 of the clusters, the AMI mass estimates are lower than both values obtained from Planck data. However the AMI and slicing function estimates are within one combined standard deviation of each other for 31 clusters. We also generate cluster simulations based on the slicing-function mass estimates, and analyse them in the same way as we did the real AMI data. We find that inclusion in the simulations of radio-source confusion, CMB noise and measurable radio-sources causes AMI mass estimates to be systematically low.

The Arcminute Microkelvin Imager (AMI) telescopes located at the Mullard Radio Astronomy Observatory near Cambridge have been significantly enhanced by the implementation of a new digital correlator with 1.2 MHz spectral resolution. This system has replaced a 750-MHz resolution analogue lag-based correlator, and was designed to mitigate the effects of radio frequency interference, particularly from geostationary satellites that contaminate observations at low declinations. The upgraded instrument consists of 18 ROACH2 Field Programmable Gate Array platforms used to implement a pair of real-time FX correlators -- one for each of AMI's two arrays. The new system separates the down-converted RF baseband signal from each AMI receiver into two 2.3 GHz-wide sub-bands which are each digitized at 5-Gsps with 8 bits of precision. These digital data streams are filtered into 2048 frequency channels and cross-correlated using FPGA hardware, with a commercial 10 Gb Ethernet switch providing high-speed data interconnect. Images formed using data from the new digital correlator show over an order of magnitude improvement in dynamic range over the previous system. The ability to observe at low declinations has also been significantly improved., Comment: 11 pages, 10 figures, accepted to MNRAS