On Choosing Strong Classifiers for Predicting Line Betting Results

The themes in this blog have been bouncing around in my thoughts - in virtual and in unpublished blog form - for quite a while now. My formal qualifications are as an Econometrician but many of the models that I find myself using in MoS come from the more recent (though still surprisingly old) Machine Learning (ML) discipline, which I'd characterise as being more concerned with the predictive ability of a model than with its theoretical pedigree. (Breiman wrote a wonderful piece on this topic, entitled Statistical Modelling: The Two Cultures, back in 2005.)

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The Responsiveness of Bookmaker Prices To Winning and Losing

In this blog I'm seeking to answer a single question: how are a team's subsequent head-to-head bookmaker prices affected by the returns they've provided to head-to-head wagering on them in recent weeks? More succinctly, how much less can you expect to make wagering on recent winners and how much more on recent losers?

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More Ways to Derive Probability and Margin Predictions From Head-to-Head Prices

A couple of weeks ago, in this earlier blog, I described a general framework for deriving probability predictions from a bookmaker's head-to-head prices and then, if required, generating margin predictions from those probability predictions.

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Assessing Probability Forecasts: Beyond Brier and Log Probability Scores

Einstein once said that "No problem can be solved from the same level of consciousness that created it". In a similar spirit - but with, regrettably and demonstrably, a mere fraction of the intellect - I find that there's something deeply satisfying about discovering that an approach to a problem you've been using can be characterised as a special case of a more general approach.

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Modelling Miscalibration

If you're making probability assessments one of the things you almost certainly want them to be is well-calibrated, and we know both from first-hand experience and a variety of analyses here on MatterOfStats over the years that the TAB Bookmaker is all of that.

Well he is, at least, well-calibrated as far as I can tell. His actual probability assessments aren't directly available but must, instead, be inferred from his head-to-head prices and I've come up with three ways of making this inference, using an Overround-Equalising, Risk-Equalising or an LPSO-Optimising approach.

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Is That a Good Probability Score - the Brier Score Edition

In recent blogs about the Very Simple Ratings System (VSRS) I've been using as my Probability Score metric the Brier Score, which assigns scores to probability estimates on the following basis:

Brier Score = (Actual Result - Probability Assigned to Actual Result)^2

For the purposes of calculating this score the Actual Result is treated as (0,1) variable, taking on a value of 1 if the team in question wins, and a value of zero if that team, instead, loses. Lower values of the Brier Score, which can be achieved by attaching large probabilities to teams that win or, equivalently, small probabilities to teams that lose, reflect better probability estimates.

Elsewhere in MAFL I've most commonly used, rather than the Brier Score, a variant of the Log Probability Score (LPS) in which a probability assessment is scored using the following equation:

Log Probability Score = 1 + logbase2(Probability Associated with Winning team)

In contrast with the Brier Score, higher log probabilities are associated with better probability estimates.

Both the Brier Score and the Log Probability Score metrics are what are called Proper Scoring Rules, and my preference for the LPS has been largely a matter of taste rather than of empirical evidence of superior efficacy.

Because the LPS has been MAFL's probability score of choice for so long, however, I have previously written a blog about empirically assessing the relative merits of a predictor's season-average LPS result in the context of the profile of pre-game probabilities that prevailed in the season under review. Such context is important because the average LPS that a well-calibrated predictor can be expected to achieve depends on the proportion of evenly-matched and highly-mismatched games in that season. (For the maths on this please refer to that earlier blog.)


What I've not done previously is provide similar, normative data about the Brier Score. That's what this blog will address.

Adopting a methodology similar to that used in the earlier blog establishing the LPS norms, for this blog I've: 

  • Calculated the implicit bookmaker probabilities (using the Risk-Equalising approach) for all games in the 2006 to 2013 period
  • Assumed that the predictor to be simulated assigns probabilities to games as if making a random selection from a Normal distribution with mean equal to the true probability - as assessed in the step above - plus some bias between -10% and +10% points, and with some standard deviation (sigma) in the range 1% to 10% points. Probability assessments that fall outside the (0.01, 0.99) range are clipped. Better tipsters are those with smaller (in absolute terms) bias and smaller sigma.
  • For each of the simulated (bias, sigma) pairs, simulated 1,000 seasons with the true probabilities for every game drawn from the empirical implicit bookmaker probabilities for a specific season.

Before I reveal the results for the first set of simulations let me first report on the season-by-season profile of implicit bookmaker probabilities, based on my TAB Sportsbet data.

The black bars reflect the number of games for which the home team's implicit home team probability fell into the bin-range recorded in the x-axis, and the blue lines map out the smoothed probability density of that same data. These blue lines highlight the similarity in terms of the profile of home team probabilities of the last three seasons. In these three years we've seen quite high numbers of short-priced (ie high probability) home team favourites and few - though not as few as in some other years - long-shot home teams. 

Seasons 2008, 2009 and 2010 saw a more even spread of home team probabilities and fewer extremes of probability at either end, though home team favourites still comfortably outnumbered home team underdogs. Seasons 2006 and 2007 were different again, with 2006 exhibiting some similarities to the 2008 to 2010 period, but with 2007 standing alone as a season with a much larger proportion of contests pitting relatively evenly-matched tips. That characteristic makes prediction more difficult, which we'd expect to be reflected in expected probability scores.

So, with a view to assessing the typical range of Brier Scores under the most diverse sets of conditions, I ran the simulation steps described above once using the home team probability distribution from 2013, and once using the distribution from 2007.


Here, firstly, are the results for all (bias, sigma) pairs, each simulated for 1,000 seasons that look like 2013.

As we'd expect, the best average Brier Scores are achieved by a tipster with zero bias and the minimum, 1% standard deviation. Such a tipster could expect to achieve an average Brier Score of about 0.167 in seasons like 2013.

For a given standard deviation, the further is the bias from zero the poorer (higher) the expected Brier Score and, for a given bias, the larger the standard deviation the poorer the expected Brier Score as well. So, for example, we can see from the graph that an unbiased tipster with a 5% point standard deviation should expect to record an average Brier Score of about 0.175.

Using Eureqa to fit an equation to the Brier Score data for all 210 simulated (bias, sigma) pairs produces the following approximation:

Expected Brier Score = 0.168 + 0.89 x Bias^2 + 0.87 x Sigma^2

This equation, which explains 98% of the variability in the average Brier Scores across the 210 combinations, suggests that the Brier Score of a tipster is about equally harmed by equivalent changes in percentage point terms in bias and variance (ie sigma squared). Every 1% point change in squared bias or in variance adds about 0.09 to the expected Brier Score.

Next, we simulate Brier Score outcomes for seasons that look like 2007 and obtain the following picture:

The general shape of the relationships shown here are virtually identical to those we saw when using the 2013 data, but the expected Brier Score values are significantly higher.

Now, an unbiased tipster with a 1% point standard deviation can expect to register a Brier Score of about 0.210 per game (up from 0.167), while one with a 5% point standard deviation can expect to return a Brier Score of about 0.212 (up from 0.175).

Eureqa now offers the following equation to explain the results for the 210 (bias, sigma) pairs:

Expected Brier Score = 0.210 + 0.98 x Bias^2 + 0.94 x Sigma^2

This equation explains 99% of the variability in average Brier Scores across the 210 combinations and, when compared with the earlier equation, suggests that: 

  • A perfect tipster - that is one with zero bias and zero variance - would achieve a Brier Score of about 0.210 in seasons like 2007 and of 0.168 in seasons like 2013
  • Additional bias and variability in a tipster's predictions are punished more in absolute terms in seasons like 2007 than in seasons like 2013. This is evidenced by the larger coefficients on the bias and variance terms in the equation for 2007 compared to those for 2013.

In seasons in which probability estimation is harder - that is, in seasons full of contests pitting evenly-matched teams against one another - Brier Scores will tend to do a better job of differentiating weak from strong predictors. 


Though I have performed simulations to determine empirical norms for the LPS metric before, I included this metric in the current round of simulations as well. Electrons are cheap.

Here are the curves for simulations of LPS for the 2013-like seasons.

Eureqa suggests that the relationship between expected LPS, bias and variance is, like that between Brier Score, bias and variance, quadratic in nature, though here the curves are concave rather than convex. We get:

Expected LPS = 0.271 - 4.68 x Bias^2 - 4.71 x Sigma^2

This equation explains 99% of the variability in average LPSs observed across the 210 combinations of bias and sigma.

Finally, simulating using 2007-like seasons gives us this picture.

Again we find that the shape when using the 2007 data is the same as that when using the 2013 data, but the absolute scores are poorer (which here means lower).

Eureqa now offers up this equation:

Expected LPS = 0.127 - 4.17 x Bias^2 - 4.35 x Sigma^2

This equation accounts for 97% of the total variability in average LPS across the 210 simulated pairs of bias and sigma and suggests that expected LPSs in seasons like 2007 are less sensitive to changes in bias and variance than are expected LPSs in seasons like 2013. This is contrary to the result we found for expected Brier Scores, which were more sensitive to changes in bias and variance in seasons like 2007 than in seasons like 2013.

In more challenging predictive environments, therefore, differences in predictive ability as measured by different biases and variances, are likely to result in larger absolute differences in Brier Scores than differences in LPSs.


We now have some bases on which to make normative judgements about Brier Scores and Log Probability Scores, though these judgements require some knowledge about the underlying distribution of true home team probabilities.

If 2014 is similar to the three seasons that have preceded it then a "good" probability predictor should produce an average Brier Score of about 0.170 to 0.175, and an average LPS of about 0.230 to 0.260. In 2013, the three bookmaker-derived Probability Predictors all finished the season with average LPS' of about 0.260.

[EDIT : It's actually not difficult to derive the following relationship theoretically for a forecaster whose predictions are 0 or 1 with fixed probability and independent of the actual outcome

Expected Brier Score = True Home Probability x (1 - True Home Probability) + Bias^2 + Sigma^2

The proof appears in the image at left.

(Click on the image for a larger version.)

Now the fitted equations for Expected Brier Scores above have coefficients on Bias and Sigma that are less than 1 mostly due, I think, to the effects of probability truncation, which tend to improve (ie lower) Brier Scores for extreme probabilities. There might also be some contribution from the fact that I've modelled the forecasts using a Normal rather than a Binomial distribution.

Deriving a similar equation theoretically rather than empirically for the Expected LPS of a contest is a far more complicated endeavour ...]

Bookmaker Implicit Probabilities: Empirical Value of the Risk-Equalising Approach

A few blogs back I developed the idea that bookmakers might embed overround in each team's price not equally but instead such that the resulting head-to-head market prices provide insurance for a fixed (in percentage point terms) calibration error of equivalent size for both teams. Since then I've made only passing comment about the empirical superiority of this approach (which I've called the Risk-Equalising Approach) relative to the previous approach (which I've called the Overround-Equalising Approach).
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A Well-Calibrated Model

It's nice to come up with a new twist on an old idea. This year, in reviewing the relative advantages and disadvantages conferred on each team by the draw, I want to do it a little differently. Specifically, I want to estimate these effects by measuring the proportion of games that I expect each team will win given their actual draw compared to the proportion I'd expect them to win if they played every team twice (yes, that hoary old chestnut in a different guise - that isn't the 'new' bit).
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Line Fund Profitability and Probability Scores

Over on the Simulations blog as part of a more general investigation into the dynamics of the contest between punter and bookmaker in head-to-head wagering I've looked at the relationship between the probability score attained by the Head-to-Head Fund in each season and its profitability. What I found, among other things, was that the Fund's profitability was related not to the absolute probability score of the Fund algorithm, but to its probability score relative to the bookmaker's.
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The Calibration of the Head-to-Head Fund Algorithm

In the previous blog we considered the logarithmic probability score on ProPred, WinPred and the TAB bookie and found that the TAB bookie was the best calibrated of the three and that relative tipping performance was somewhat unrelated to relative probability scores. For the Head-to-Head Fund, whose job in life is to make money, the key question is to what extent do its probability scores relative to the TAB bookie's shed light on its money-making prowess.
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Assessing ProPred's, WinPred's and the Bookie's Probability Forecasts

Almost 12 months ago, in this blog, I introduced the topic of probability scoring as a basis on which to assess the forecasting performance of a probabilistic tipster. Unfortunately, I used it for the remainder of last season as a means of assessing the ill-fated HELP algorithm, which didn't so much need a probability score to measure its awfullness as it did a stenchometer. As a consequence I think I'd mentally tainted the measure, but it deserves another run with another algorithm.
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Why It Matters Which Team Wins

In conversation - and in interrogation, come to think of it - the key to getting a good answer is often in the framing of the question.

So too in statistical modelling, where one common method for asking a slightly different question of the data is to take the variables you have and transform them.

Consider for example the following results for four binary logits, each built to provide an answer to the question 'Under what circumstances does the team with the higher MARS Rating tend to win?'.
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In-Running Wagering: What's the Best Strategy?

With services such as Betfair now offering in-running wagering opportunities, the ability to accurately assess a team's chances of victory at any given point in a game is now of considerable commercial value. Imagine, for example, that your team, who are at home, lead by 18 points at the first change. Would a wager on them at $1.40 be advised?
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