Introduction Recall from the previous post that the Hare growth parameter undergoes Brownian motion so that the further into the future we go, the less certain we are about it. In order to ensure that this parameter remains positive, let’s model the log of it to be Brownian motion. where the final equation is a … Continue reading Modelling an…
#bayesian
16 posts
4 Jul 2016
26 Jun 2016
Introduction In the 1920s, Lotka (1909) and Volterra (1926) developed a model of a very simple predator-prey ecosystem. Although simple, it turns out that the Canadian lynx and showshoe hare are well represented by such a model. Furthermore, the Hudson Bay Company kept records of how many pelts of each species were trapped for almost … Continue reading Ecology, Dynamical…
1 May 2016
Introduction This is a bit different from my usual posts (well apart from my write up of hacking at Odessa) in that it is a log of how I managed to get LibBi (Library for Bayesian Inference) to run on my MacBook and then not totally satisfactorily (as you will see if you read on). … Continue reading Fun with…
17 Apr 2016
Introduction In their paper Betancourt et al. (2014), the authors give a corollary which starts with the phrase “Because the manifold is paracompact”. It wasn’t immediately clear why the manifold was paracompact or indeed what paracompactness meant although it was clearly something like compactness which means that every cover has a finite sub-cover. It turns … Continue reading Every Manifold…
20 Jan 2016
Introduction The equation of motion for a pendulum of unit length subject to Gaussian white noise is We can discretize this via the usual Euler method where and The explanation of the precise form of the covariance matrix will be the subject of another blog post; for the purpose of exposition of forward filtering / … Continue reading Particle Smoothing
15 Jan 2016
Introduction The equation of motion for a pendulum of unit length subject to Gaussian white noise is We can discretize this via the usual Euler method where and The explanation of the precise form of the covariance matrix will be the subject of another blog post; for the purpose of exposition of using Stan and, … Continue reading Inferring Parameters…
6 Dec 2015
Introduction Let be a (hidden) Markov process. By hidden, we mean that we are not able to observe it. And let be an observable Markov process such that That is the observations are conditionally independent given the state of the hidden process. As an example let us take the one given in Särkkä (2013) where … Continue reading Naive Particle…
7 Jun 2015
Here’s the same analysis of estimating population growth using Stan. data { int<lower=0> N; // number of observations vector[N] y; // observed population } parameters { real r; } model { real k; real p0; real deltaT; real sigma; real mu0; real sigma0; vector[N] p; k <- 1.0; p0 <- 0.1; deltaT <- 0.0005; sigma … Continue reading Population Growth…
6 Jun 2015
Introduction Let us see if we can estimate the parameter for population growth using MCMC in the example in which we used Kalman filtering. We recall the model. And we are allowed to sample at regular intervals In other words , where is known so the likelihood is Let us assume a prior of then … Continue reading Population Growth…
11 Mar 2015
Introduction Simple models for e.g. financial option pricing assume that the volatility of an index or a stock is constant, see here for example. However, simple observation of time series show that this is not the case; if it were then the log returns would be white noise One approach which addresses this, GARCH (Generalised AutoRegressive … Continue reading Stochastic…
9 Sept 2014
Summary An extended Kalman filter in Haskell using type level literals and automatic differentiation to provide some guarantees of correctness. Population Growth Suppose we wish to model population growth of bees via the logistic equation We assume the growth rate is unknown and drawn from a normal distribution but the carrying capacity is known and … Continue reading Fun with…
23 Aug 2014
Importance Sampling Suppose we have an random variable with pdf and we wish to find its second moment numerically. However, the random-fu package does not support sampling from such as distribution. We notice that So we can sample from and evaluate > {-# OPTIONS_GHC -Wall #-} > {-# OPTIONS_GHC -fno-warn-name-shadowing #-} > {-# OPTIONS_GHC -fno-warn-type-defaults … Continue reading Importance Sampling
6 Aug 2014
Introduction Suppose we have particle moving in at constant velocity in 1 dimension, where the velocity is sampled from a distribution. We can observe the position of the particle at fixed intervals and we wish to estimate its initial velocity. For generality, let us assume that the positions and the velocities can be perturbed at … Continue reading Fun with…
19 Jul 2014
Suppose we wish to estimate the mean of a sample drawn from a normal distribution. In the Bayesian approach, we know the prior distribution for the mean (it could be a non-informative prior) and then we update this with our observations to create the posterior, the latter giving us improved information about the distribution of … Continue reading Fun with…
15 Jun 2014
This is really intended as a draft chapter for our book. Given the diverse natures of the intended intended audiences, it is probably a bit light on explanation of the Haskell (use of monad transformers) for those with a background in numerical methods. It is hoped that the explanation of the mathematics is adequate for … Continue reading Gibbs Sampling…
9 Apr 2014
Introduction It’s possible to Gibbs sampling in most languages and since I am doing some work in R and some work in Haskell, I thought I’d present a simple example in both languages: estimating the mean from a normal distribution with unknown mean and variance. Although one can do Gibbs sampling directly in R, it … Continue reading Gibbs Sampling…