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We have a matrix, $\mathbf{X}$, of probability distributions between 6 different results, so each row $\mathbf{x}_i$ sums to 1. We want to perform dimension reduction so that each row is a linear interpolation of two components, $\mathbf{f}$ and $\mathbf{g}$, which are also probability distributions. In other words, each row $\mathbf{x}_i$ has a latent variable $c_i$ so that $\mathbf{x}_i \sim c_i \mathbf{f} + (1 - c_i) \mathbf{g}$. Furthermore, we have the following constraints:

  • Certain probabilities in the components are 0:
    • $\mathbf{f}$ is of the form $(f_0, 0, f_2, f_3, f_4, f_5)$.
    • $\mathbf{g}$ is of the form $(0, g_1, g_2, g_3, g_4, 0)$.
  • As probability distributions, $\mathbf{f}$ and $\mathbf{g}$ also sum to 1.

In addition, some results are more important than others to estimate accurately, so we would like to weight the least squares error: Let the estimate of row $x_i$ be $\hat{x}_i = c_i \mathbf{f} + (1 - c_i) \mathbf{g}$. We would like to minimize the weighted sum of squared residual, $$\sum_{j=0}^{5} w_j (x_j - \hat{x}_j)^2.$$ I'm currently working in Python but am open to other tools to solve this problem. Thank you.


Context: For the game of Blaseball, for each player, we have counts of 6 events when the ball is in-play: flyout, ground_out, single, double, triple, and home_run. We hypothesize a generative model where first either a fly or a grounder occurs, and then the corresponding probability distribution, $\mathbf{f}$ or $\mathbf{g}$, is used to generate the observed event. Each player has a latent variable that describes their probability of selecting either $\mathbf{f}$ or $\mathbf{g}$. The constraint of 0 in certain indices of $\mathbf{f}$ and $\mathbf{g}$ is the assumption that certain generator-observation pairs are impossible, i.e., a fly cannot result in a ground_out, a grounder cannot result in a flyout or home_run.

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    $\begingroup$ A natural tool is a probabilistic programming language for statistical inference, such as Stan. Choose some suitable probability distribution over probability vectors $D_f$ and $D_g$; an example is the dirichlet-distribution. To model the constraint, set the distribution's parameters to values that assign minuscule probability to the values you want to be 0. $\endgroup$
    – Sycorax
    Commented May 12 at 4:42
  • $\begingroup$ would you mind adding to your description why you want to make these constraints, and what your overall purpose is? $\endgroup$
    – seanv507
    Commented May 12 at 8:47
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    $\begingroup$ I have added some context to explain the constraints, thanks $\endgroup$
    – jgf1123
    Commented May 13 at 18:43

1 Answer 1

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i would think this could be done relatively easily using an unconstrained nonlinear least squares optimiser such as levenberg marquardt

you can turn your probability constraints on f_j, into unconstrained variables,z_j, using a 'multinomial transformation'

model $f_j =\exp(z_j)/(1+\sum _k exp(z_k))$

where the sum is from 0 to 4, and f_5 is calculated as 1 - that sum.

and similarly for the gs and cs


scipy provides a non linear constrained optimisation function https://docs.scipy.org/doc/scipy-1.13.0/reference/generated/scipy.optimize.least_squares.html

so you just constrain f_0 to f_4 to lie between 0 and 1 and f_5 is computed as 1- sum.

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