Model implementation technical details

Introduction

In this post I will present the details of the resampling filter that was presented in my previous post. More specifically, I will describe the Gibbs sampling algorithm #4 as it was described in Radford M. Neal's paper. This was the basic algorithm that was used in the previous post to augment the resampling process of the FastSLAM algorithm. In Neal's paper the information is presented in a nice and neat but also very abstract manner, so it's difficult for an uninitiated reader to follow all the details. Therefore, information on this post will also take elements from this paper from Herman Kamper as well as this from Xiaodong Yu. In the rest post I will first describe the algorithm as presented by Neal and then I will talk about it's details more explicitly. Finally I will present how these are applied to the aforementioned filter. Again some background on DPMM and MCMC methods is assumed.

Initial description of the collapsed CRP algorithm.

The description of the algorithm taken as is from Neal's paper is the following:

Finally, in a conjugate context, we can often intergrate analytically over the \(\begin{equation} \phi_{c} \end{equation}\), eliminating them from the algorithm. The state of the Markov chain then consists only of the \(\begin{equation} c_{i} \end{equation}\) which we update by Gibbs sampling using the following conditional probabilitites.

\begin{cases}if\  c=c_{j}\ for\ some\ j\neq i:\ P(c_i=c)= b\frac{n_{-i,c}}{n-1+a}\int_{}{}F(y_i,\phi)dH_{-i,c}(\phi) \\\ \ \ \ \ \ \ \ \ \ P(c_i\neq c_j\ for\ all\ i \neq\ j|c_{-i},y_i)=b\frac{a}{n-1+a}\int_{}{}F(y_i,\phi)dG_0(\phi)\end{cases}

Here, \(\begin{equation} H_{-i,c} \end{equation}\) is the posterior distribution of \(\begin{equation} \phi \end{equation}\) based on the prior \(\begin{equation} G_0 \end{equation}\) and all observations \(\begin{equation} y_j \end{equation}\) for which \(\begin{equation} j \neq i \end{equation}\) and \(\begin{equation} c_j =c \end{equation}\).

The third Gibbs sampling method can be summarized as follows:

Algorithm 3: Collapsed Gibbs sampling

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Let the state of the Markov chain consist of \(\begin{equation} c_1,...,c_n \end{equation}\). Repeatedly sample as follows:

• For \(\begin{equation} i=1,...n \end{equation}\) Draw a new value from \(\begin{equation} c_i|c_{-i},y_i \end{equation}\) as defined by the previous equation.

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Yes, that's all, no joking here.

A resampling example using matlab.

Introduction

In the previous post, a brief introduction on the Dirichlet distribution, Dirichlet process as well as finite and infinite mixture models was given. Furthermore, methods that can be used to perform inference on such models were enumerated. Finally the research question was revisited and the problem, as well as the approach towards solving it were given.
Before continuing to ROS and a low-level implementation of the model, I wanted to have a working example on matlab taking advantage of its nice vectorized functions.

Current setup

The current setup on matlab is now the following:

• A matalb-based FastSLAM implementation by Grace Lee that was already used in post #1 can be found here
• A matlab-based Gibbs sampler by François Caron can be found here

The particle positions will be used as input to the Gibbs sampler who will cluster the particles as a mixture of Gaussian distributions. The base distribution that will be used as a prior to the Dirichlet process on the sampler will be a Normal-Inverse-Wishart distribution and was chosen due to its conjugacy to the Gaussian. These mixtures output by the sampler will then be used to train a Hierarchical Dirichlet process with the cluster significance proportional to the particle weights that are classified within the cluster and particle significance proportional to the given particle weight.
An example on the single iteration on the resampling step is then the following:

Model description-Warning long post!

Post #3

Introduction

In this post I will describe the basic structure of the filter that I plan on implementing in the resampling step of the FastSLAM .The post is structured as follows:
First a brief introduction regarding Dirichlet processes as well as the basic finite and infinite mixture models using such processes are given. Then the basic inference methods on finite and infinite mixture models are presented. Finally the relation that connects such models to my research topic is analysed.
Reading material regarding DP's can be found here and here. Wikipedia pages have useful but long articles regarding the Dirichlet Distribution and Dirichlet Process and their respective properties(exchangeability, aggregation etc) .

The Dirichlet process, a fast introduction.

Dirichlet distribution: Let :

Be a random pmf.  In addition, suppose that