Archive for normalisation

a case for Bayesian deep learnin

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , on September 30, 2020 by xi'an

Andrew Wilson wrote a piece about Bayesian deep learning last winter. Which I just read. It starts with the (posterior) predictive distribution being the core of Bayesian model evaluation or of model (epistemic) uncertainty.

“On the other hand, a flat prior may have a major effect on marginalization.”

Interesting sentence, as, from my viewpoint, using a flat prior is a no-no when running model evaluation since the marginal likelihood (or evidence) is no longer a probability density. (Check Lindley-Jeffreys’ paradox in this tribune.) The author then goes for an argument in favour of a Bayesian approach to deep neural networks for the reason that data cannot be informative on every parameter in the network, which should then be integrated out wrt a prior. He also draws a parallel between deep ensemble learning, where random initialisations produce different fits, with posterior distributions, although the equivalent to the prior distribution in an optimisation exercise is somewhat vague.

“…we do not need samples from a posterior, or even a faithful approximation to the posterior. We need to evaluate the posterior in places that will make the greatest contributions to the [posterior predictive].”

The paper also contains an interesting point distinguishing between priors over parameters and priors over functions, ony the later mattering for prediction. Which must be structured enough to compensate for the lack of data information about most aspects of the functions. The paper further discusses uninformative priors (over the parameters) in the O’Bayes sense as a default way to select priors. It is however unclear to me how this discussion accounts for the problems met in high dimensions by standard uninformative solutions. More aggressively penalising priors may be needed, as those found in high dimension variable selection. As in e.g. the 10⁷ dimensional space mentioned in the paper. Interesting read all in all!

probabilities larger than one…

Posted in Statistics with tags , , , , , , on November 9, 2017 by xi'an

asymptotically exact inference in likelihood-free models [a reply from the authors]

Posted in R, Statistics with tags , , , , , , , , , , , , , , , , , on December 1, 2016 by xi'an

[Following my post of lastTuesday, Matt Graham commented on the paper with force détails. Here are those comments. A nicer HTML version of the Markdown reply below is also available on Github.]

Thanks for the comments on the paper!

A few additional replies to augment what Amos wrote:

This however sounds somewhat intense in that it involves a quasi-Newton resolution at each step.

The method is definitely computationally expensive. If the constraint function is of the form of a function from an M-dimensional space to an N-dimensional space, with MN, for large N the dominant costs at each timestep are usually the constraint Jacobian (c/u) evaluation (with reverse-mode automatic differentiation this can be evaluated at a cost of O(N) generator / constraint evaluations) and Cholesky decomposition of the Jacobian product (c/u)(c/u) with O(N³) cost (though in many cases e.g. i.i.d. or Markovian simulated data, structure in the generator Jacobian can be exploited to give a significantly reduced cost). Each inner Quasi-Newton update involves a pair of triangular solve operations which have a O(N²) cost, two matrix-vector multiplications with O(MN) cost, and a single constraint / generator function evaluation; the number of Quasi-Newton updates required for convergence in the numerical experiments tended to be much less than N hence the Quasi-Newton iteration tended not to be the main cost.

The high computation cost per update is traded off however with often being able to make much larger proposed moves in high-dimensional state spaces with a high chance of acceptance compared to ABC MCMC approaches. Even in the relatively small Lotka-Volterra example we provide which has an input dimension of 104 (four inputs which map to ‘parameters’, and 100 inputs which map to ‘noise’ variables), the ABC MCMC chains using the coarse ABC kernel radius ϵ=100 with comparably very cheap updates were significantly less efficient in terms of effective sample size / computation time than the proposed constrained HMC approach. This was in large part due to the elliptical slice sampling updates in the ABC MCMC chains generally collapsing down to very small moves even for this relatively coarse ϵ. Performance was even worse using non-adaptive ABC MCMC methods and for smaller ϵ, and for higher input dimensions (e.g. using a longer sequence with correspondingly more random inputs) the comparison becomes even more favourable for the constrained HMC approach. Continue reading