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Showing posts from February, 2017

Kronecker Product for Matrices with Special Structure Using LinearOperators

I have been working with very large matrices for my research lately, and some of them are sparse, while others are dense but have some special structure that can be exploited to represent them efficiently.  I have also been constructing enormous matrices from these smaller building blocks using the Kronecker product.  Both numpy and scipy have functions built in to compute the Kronecker product between two matrices, but they need the input matrices to be explicitly represented (as a dense numpy array or a sparse scipy matrix), and so they are not compatible with the special-structure matrices that I've been dealing with.  I recently solved this problem, and that is the topic of this blog post.
An Example As an example to make things more concrete, consider the two matrices shown below:
$$ A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}  \qquad B = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 &…

Modeling and Predicting the Outcome of a Sequence of Games

Image
Here's something I though of today that I think is pretty interesting. Consider the following toy problem:
Suppose Alice and Bob play a sequence of games, and the probability that Alice wins game $i$ is given by $p_i$. Further, assume $p_1 \sim U[0,1]$ is uniform and that $ p_{i+1} \sim Beta(\alpha p_i + \beta, \alpha (1-p_i) + \beta) $ for known $ (\alpha, \beta) $. We are unable to directly observe $p_i$, but we do observe the outcome of each game $ o_i $ for $ 1 \leq i \leq k $. The task is to estimate $ p_{k+1} $ given the evidence $ o_1, \dots, o_k $.   The figure below shows how Alice's win probability might evolve over time.  This particular figure was generated with $ \alpha = 1000 $ and $ \beta = 5 $.

We can model this situation with a modified Hidden Markov Model.  The graph below shows how the random variables are related in the form of a directed graphical model:

There are very efficient algorithms for doing inference and learning on Hidden Markov Models, but t…