Multimodal nested sampling: an efficient and robust alternative to MCMC methods for astronomical data analysis

In performing a Bayesian analysis of astronomical data, two difficult problems often emerge. First, in estimating the parameters of some model for the data, the resulting posterior distribution may be multimodal or exhibit pronounced (curving) degeneracies, which can cause problems for traditional MCMC sampling methods. Second, in selecting between a set of competing models, calculation of the Bayesian evidence for each model is computationally expensive. The nested sampling method introduced by Skilling (2004), has greatly reduced the computational expense of calculating evidences and also produces posterior inferences as a by-product. This method has been applied successfully in cosmological applications by Mukherjee et al. (2006), but their implementation was efficient only for unimodal distributions without pronounced degeneracies. Shaw et al. (2007), recently introduced a clustered nested sampling method which is significantly more efficient in sampling from multimodal posteriors and also determines the expectation and variance of the final evidence from a single run of the algorithm, hence providing a further increase in efficiency. In this paper, we build on the work of Shaw et al. and present three new methods for sampling and evidence evaluation from distributions that may contain multiple modes and significant degeneracies; we also present an even more efficient technique for estimating the uncertainty on the evaluated evidence. These methods lead to a further substantial improvement in sampling efficiency and robustness, and are applied to toy problems to demonstrate the accuracy and economy of the evidence calculation and parameter estimation. Finally, we discuss the use of these methods in performing Bayesian object detection in astronomical datasets.Comment: 14 pages, 11 figures, submitted to MNRAS, some major additions to the previous version in response to the referee's comment

A Bayesian approach to strong lensing modelling of galaxy clusters

In this paper, we describe a procedure for modelling strong lensing galaxy clusters with parametric methods, and to rank models quantitatively using the Bayesian evidence. We use a publicly available Markov chain Monte-Carlo (MCMC) sampler ('Bayesys'), allowing us to avoid local minima in the likelihood functions. To illustrate the power of the MCMC technique, we simulate three clusters of galaxies, each composed of a cluster-scale halo and a set of perturbing galaxy-scale subhalos. We ray-trace three light beams through each model to produce a catalogue of multiple images, and then use the MCMC sampler to recover the model parameters in the three different lensing configurations. We find that, for typical Hubble Space Telescope (HST)-quality imaging data, the total mass in the Einstein radius is recovered with ~1-5% error according to the considered lensing configuration. However, we find that the mass of the galaxies is strongly degenerated with the cluster mass when no multiple images appear in the cluster centre. The mass of the galaxies is generally recovered with a 20% error, largely due to the poorly constrained cut-off radius. Finally, we describe how to rank models quantitatively using the Bayesian evidence. We confirm the ability of strong lensing to constrain the mass profile in the central region of galaxy clusters in this way. Ultimately, such a method applied to strong lensing clusters with a very large number of multiple images may provide unique geometrical constraints on cosmology. The implementation of the MCMC sampler used in this paper has been done within the framework of the Lenstool software package, which is publicly available.Comment: Accepted to "Gravitational Lensing" Focus Issue of the New Journal of Physics (invited), 35 pages, 11 figures at reduced resolutio

Numerical Bayesian methods applied to signal processing

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Rotation, scale and translation invariant spread spectrum digital image watermarking

A digital watermark is an invisible mark embedded in a digital image which may be used for a number of different purposes including image captioning and copyright protection. This paper describes how a combination of spread spectrum encoding of the embedded message and transform-based invariants can be used for digital image watermarking. In particular, it is described how a Fourier-Mellin-based approach can be used to construct watermarks which are designed to be unaffected by any combination of rotation and scale transformations. In addition, a novel method of CDMA spread spectrum encoding is introduced which allows one to embed watermark messages of arbitrary length and which need only a secret key for decoding. The paper also describes the usefulness of Reed Solomon error-correcting codes in this scheme

Rotation, scale and translation invariant digital image watermarking

A digital watermark is an invisible mark embedded in a digital image which may be used for Copyright Protection. This paper describes how Fourier-Mellin transform-based invariants can be used for digital image watermarking. The embedded marks are designed to be unaffected by any combination of rotation, scale and translation transformations. The original image is not required for extracting the embedded mark

Watermarking methods

Watermarks. The term evokes visions of shady characters secretly beavering away in dark basements surrounded by forged $100 bills drying on clothes lines. In a digital media context, away from the traditional world of inks and paper, the same old problem remains but it relates not just to forgery but also to outright theft, because one digital copy can spawn millions of others with the single click of a mouse button. It is hardly surprising that the notion of a digital watermark has stimulated avid interest amongst artists and publishers alike. It is commonly recognized that digital watermarks must be as robust as the media in which they are embedded. For example, a rotated, cropped, and rescanned water-marked image should still be a watermarked image. However, this robustness requirement directly conflicts with the need for a digital watermark to be unobtrusive. The most effective techniques used to embed watermarks are the result of a combination of secret key-based techniques used for military communication and simple models of the human visual system. The most familiar application for digital watermarks is for copyright protection and protection of intellectual property. On its own, a watermark does not provide any legal proof of ownership. In other words, the use of a given digital watermark to protect intellectual property must be registered with a trusted third party to be of any value. Any technique for embedding robust digital watermarks must be compatible with methods for registering copyright. A watermark's resistance to intentional and unintentional degradation has been the main subject of interest in the watermarking community. The main challenges are geometric transformations such as change of proportion or simple rescaling. One watermark removal technique that is supplied on the Internet simply shifts a corner of the image. Lossy image compression such as JPEG and filtering are more easily overcome and, generally speaking, watermarks have evolved into very resistant forms. The laboratory, but will they really work in the real world