## 2023 NHL Playoff Predictions

Who will win this year’s cup?

This year I’ve been trying to learn more about probabilistic programming and Bayesian inference. Additionally, I’ve been playing around with some other programming languages, R and Julia. This week’s Riddler question at FiveThirtyEight seemed like a good opportunity to utilize these things - I did this solution in Julia, using the Turing.jl probabilistic programming library. The code for this solution can be found on my GitHub.

This solution mentioned in the featured solutions - see here.

When you started your doctorate several years ago, your astrophysics lab noticed some unusual signals coming in from deep space on a particular frequency — hydrogen times tau. After analyzing a trove of data measured at many regular intervals, you compute that you heard zero signals in 45 percent of the intervals, one signal in 38 percent of the intervals and two signals in the remaining 17 percent of the intervals.

Your research adviser suggests that it may just be random fluctuations from two sources. Each source had some fixed probability of emitting a signal that you picked up, and together those sources generated the pattern in your data.

What do you think? Was it possible for your data to have come from two random fluctuations, as your adviser suggests?

To solve this, first I simulated observations consistent with the problem statement:

- 45% 0’s
- 38% 1’s
- 17% 2’s

To achieve that, I threw random numbers (\(n=25000\)) on the interval \([0,1]\) with floating point precision. Then to achieve the proportions as designated, I set all instances on the interval \([0.00, 0.45)\) as 0, \([0.45, 0.83)\) as 1 and \([0.83, 1.00]\) as 2. This generated the following distribution of data, where the histogram is normalized to 1 in order to show rate rather than raw counts.

This histogram is consistent with the given rates, so the generated data appears valid, with 25,000 observations.

This data can be modeled using a Poisson Binomial distribution. This distribution is the convolution of \(n\) Bernoulli distributed variables, with probability \(p_1, p_2, \dots, p_n\). In our case, we have \(n=2\) where each source omits a signal with probability \(p_i\), for \(i=1,2\) during each sampling interval.

Generic Poisson Binomial distributions with \(i=2\) for various \(p_i\) values are shown in the figure below.

Given two sources emitting with some fixed probability, this is the type of distribution I would expect. The distribution is bounded from 0-2, corresponding to neither source, one source, or both sources emitting during a sampling period. If both \(p\) values are low, the sources aren’t emitting frequently, so we expect many samplings with 0 readings. If both \(p\) values are high, we expect many cases in which we receive signals from both, so many 2 readings. Anywhere in the middle, we run into a high degree of one or the other emitting a signal, so 1 is most common. This type of behavior we observe in these generic distributions, so this model seems to fit the problem.

Formalizing the model,

\[\color{white}{y_i \sim \text{PoissonBinomial}(\mathbf{p})}\] \[\color{white}{p_i \sim \text{Uniform}(0,1)}\]Flat priors are used for \(p_1\) and \(p_2\). Given our data, we see 0 is the most common value, so most likely both \(p\) values will be low, below 0.5. However, since many observations were simulated (25,000), there’s flexibility to have less informative priors. With so many observations, the data will be the dominant factor in shaping the posterior.

The flat prior, \(\text{Uniform}(0,1)\), does provide information to the model in terms of bounds - on each time interval, each source has between a 0% and 100% chance of emitting. We don’t permit for negative or greater than 100% probabilities, since those don’t make much physical sense.

The Turing.jl library is used to construct this model. 9,000 posterior samples are run on 4 chains after 1,000 sample warm-up. This is definitely overkill, but nothing else is running on my computer overnight.

The output after fitting gives a mean of 0.358 for both \(p_i\) values, with a standard deviation of 0.006. There’s no reason the model ought to prefer one \(p\) over another, so it makes sense their mean and standard deviations are the same. The 95% credible intervals are from 0.349 to 0.368, a bit narrower than I would have expected. This means that to see the pattern we’ve gotten with two sources, both must have a probability of around 0.358 of emitting per sampling period.

Inspecting the chains and posteriors to make sure convergence is good:

Note: p_a and p_b are used in lieu of \(p_1\) and \(p_2\), by choice of personal code reading preference

This all looks normal, the fit appears to have worked well. Further, the r_hat values are 1.0001 and 1.0003, indicating good fits. The distributions appear roughly normal, centered at the above mean of 0.358 for both. Probably the most interesting check is how the probabilities covary.

This solution approach searches for possible parameter values that can realize the observations. Meaning, to get the same observations, as one \(p\) value increases, the other \(p\) value ought to decrease. In other words, if one source is emitting signals more often, to get the same realized data, the other source must emit less often. Thus, we expect a negative correlation between the two emission probability parameters.

In fact, the parameters do have a negative correlation, -0.697.

Getting back to the question posed:

Your research adviser suggests that it may just be random fluctuations from two sources. Each source had some fixed probability of emitting a signal that you picked up, and together those sources generated the pattern in your data.

What do you think? Was it possible for your data to have come from two random fluctuations, as your adviser suggests?

This analysis shows, yes, this is definitely possible. If it were true, we would expect both sources to have between a 34.7% and 36.8% probability of emitting a signal each sampling period, the 95% credible interval. The mean emission probability per sampling period we expect from either source 35.8%, with a standard deviation of 0.6%.

With so many posterior samples, its interesting from a sheer curiosity standpoint to look at the most extreme values that could result in this dataset. If one is emitting with the highest possible probability to realize this data, what is that probability? Subsequently, how infrequent would the other have to emit in this scenario?

The highest emission probability consistent with our data was found to be \(p = 0.382\), where the corresponding probability from the second source is \(p = 0.344\) (outside the lower limits of the 95% credible interval, expected due to covariance).

The lowest emission probability was found to be \(p=0.335\), with a corresponding probability from the second source of \(p=0.378\) (outside upper limits of 95% credible credible interval).

As mentioned, these are the most extreme values, so very, very unlikely to be realized, and we should lend effectively no belief to values below \(p=0.33\) or above \(p=0.39\).

- Posted August 27, 2020

After submission, I realized this solution isn’t 100% correct. The quickest red flag that I should have considered was plotting the most observed data proportions against a Poisson Binomial function using the most likely parameter values.

The given proportions in the problem aren’t anywhere close to the Poisson Binomial distribution these p values. Whoops. This should be causing alarm bells that I previously missed, and it caused me to backtrack and think some more about this problem. Like I mentioned above, these are new tools and analysis approaches that I’m currently developing, so I’m taking this error as a learning experience.

This back-of-the-envelope check isn’t 100% correct though - it’s ignoring the fact that the two values covary and just picking the most probable values. In order to appropriately compare our model to observed data, I quickly re-implemented the model to get the posterior. The posterior contained explicitly covarying sets of values for p_a and p_b, so using those sets of values, I generated sets of observations equal to our original dataset (n=25000).

Here, the blue histogram shows the expected central value given the parameter values from sampling our posterior. The green shading around the top indicates the 2σ band (95% coverage), and the black is the original data, with Poisson errors.

Given this, I should cycle back to my answer. This model would need a 23.4σ deviation in order to cover the data proportions we observe, this gives us incredible certainty that it isn’t generating our data. Reminder that 5σ equates to 99.99994% confidence, so 23.4σ is effectively “not likely to happen by the inevitable heat death of the universe” territory, so maybe, in this case, you shouldn’t be as trusting of your advisor as I originally thought.

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