## Hidden Markov Models (part II): forward-backward algorithm for marginal conditional probability of the states

(in the same series HMM (part I): recurrence equations for filtering and prediction)

Consider a Hidden Markov Model (HMM) with hidden states $x_t$ (for $t \in {1, 2, \cdots, T}$), initial probability $p(x_1)$, observed states $y_t$, transition probability $p(x_t|x_{t-1})$ and observation model $p(y_t|x_t)$. This model can be factorized as
$p(x_{1:T},y_{1:T}) = p(y_1|x_1)p(x_1)\prod_{t=2}^{t=T}p(y_t|x_t)p(x_t|x_{t-1})$. We will use the notation $X=x_{1:T}$ to represent the set $X=\{x_1,x_2,\cdots,x_T\}$.
In this post we will present the details of the method to find the smoothing distribution $p(x_t|y_{1:T})$ of a HMM, given a set of observations $y_{1:T}$:
Our starting point is the marginal probability $p(x_t|y_{1:T})$ of $x_t$ given all the observations $y_{1:T}$.

\begin{aligned} p(x_t|y_{1:T}) &= \frac{p(x_t,y_{1:T})}{p(y_{1:T})} \\ &= \frac{p(x_t,y_{1:t},y_{(t+1):T})}{p(y_{1:T})}\\ &= \underbrace{p(y_{(t+1):T}|x_t)}_{\beta_t(x_t)}\underbrace{p(x_t,y_{1:t})}_{\alpha_t(x_t)}\frac{1}{p(y_{1:T})} \\ &= \frac{\alpha_t(x_t) \beta_t(x_t)}{p(y_{1:T})} \end{aligned}

## Script for searching and web crawling Springer Books

So, some of you may have heard about the free e-books frenzy at Springer (unfortunately, not working any more ). I jumped on the train of people trying to find the goodies on the website and ended up developing some tools to help me with that. Even after the freebies ended I kept on improving this tool because it is useful also in my day-to-day job of searching for bibliography and storing those PDF’s. So this script can basically navigate through a search result (passed to the script as a link to the search result) and download all books available (even if it consist of many pages results). This is also useful for downloading all books from a series (for example, springer series in statistics). In the future I intend to update the script including download of chapters of books and research articles. This script is specially useful if your university have access to Springer Books. (full code in https://gist.github.com/zehsilva/d890837cdfd4a20ee8a5#file-download-books-from-rss-springer-py) Continue reading “Script for searching and web crawling Springer Books”

## Embedding GIF animations in IPython notebook

A couple of days ago I was thinking about the possibility of generating GIF animations using matplotlib and visualizing this animations directly in IPython notebooks. So I did a quick search to find possible solutions, and found one solution to embed videos, and other to convert matplotlib animations to gif, so I combined both solution in a third solution converting the animation to GIF using imagemagick (you will need to have it installed in your computer), enconding the resulting file in a base64 string and embedding this in a img tag. Using IPython.display HTML class this resulting string will be displayed in the notebook (and saved with the notebook).

I will briefly explain the elements of my solution.