A generative Markov model for bowling scores

Douglas VanDerwerken; Franklin Kenter

doi:10.1515/jqas-2017-0081

Article

A generative Markov model for bowling scores

Douglas VanDerwerken and Franklin Kenter

Published/Copyright: September 5, 2018

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From the journal Journal of Quantitative Analysis in Sports Volume 14 Issue 4

Abstract

We create a data-driven Markov model for generating 10-pin bowling scores from the Professional Bowlers Association. The model incorporates insights from the hot hand literature and makes use of Bayesian shrinkage. For realistic sample sizes, the proposed approach is superior to modeling via the empirical distribution. Investigation of player-specific model components allows for a richer comparison of players than is possible using raw game scores alone. An additional feature of the model is that it can be used for in-game prediction.

Keywords: Bayesian hierarchical model; bowling; hot hand; Markov model

Acknowledgement

The authors would like to thank Ben Blatt and Yaari Gur for providing us with the data they used in their analyses. The data set we ended up using for this paper was the one provided by Mr. Blatt.

8 Appendix

Table 5:

Posterior means for T_k’s with Bayesian shrinkage.

	−−−−	−−−	−−	−	X	XX	XXX	XXXX
Amleto Monacelli	0.559	0.570	0.572	0.584	0.589	0.602	0.616	0.630
Bill O’Neill	0.545	0.604	0.591	0.612	0.603	0.616	0.620	0.636
Bob Learn Jr.	0.542	0.584	0.587	0.607	0.607	0.606	0.618	0.632
Brad Angelo	0.574	0.605	0.605	0.591	0.601	0.610	0.617	0.634
Brian Himmler	0.530	0.571	0.591	0.602	0.599	0.607	0.622	0.641
Brian LeClair	0.513	0.606	0.616	0.597	0.580	0.596	0.613	0.623
Brian Voss	0.534	0.586	0.618	0.620	0.615	0.611	0.621	0.655
Brian Kretzer	0.558	0.564	0.560	0.573	0.590	0.586	0.609	0.616
Chris Barnes	0.565	0.594	0.570	0.622	0.624	0.635	0.624	0.657
Chris Collins	0.558	0.568	0.567	0.603	0.599	0.603	0.614	0.657
Chris Johnson	0.545	0.541	0.559	0.607	0.598	0.609	0.620	0.622
Chris Loschetter	0.503	0.594	0.576	0.593	0.590	0.623	0.617	0.638
Danny Wiseman	0.534	0.603	0.593	0.606	0.608	0.619	0.618	0.632
Dave D’Entremont	0.548	0.551	0.533	0.586	0.583	0.593	0.618	0.627
David Traber	0.551	0.581	0.645	0.621	0.608	0.603	0.625	0.645
Dick Allen	0.549	0.569	0.582	0.595	0.594	0.619	0.616	0.658
Doug Kent	0.544	0.583	0.561	0.605	0.579	0.621	0.605	0.643
Eugene McCune	0.562	0.590	0.575	0.603	0.587	0.621	0.623	0.649
Hugh Miller	0.547	0.577	0.563	0.607	0.595	0.602	0.618	0.629
Jack Jurek	0.551	0.577	0.591	0.600	0.608	0.597	0.610	0.637
Jason Couch	0.560	0.568	0.642	0.626	0.616	0.611	0.624	0.647
Jason Hurd	0.531	0.595	0.613	0.587	0.600	0.604	0.613	0.634
Jeff Carter	0.538	0.573	0.573	0.608	0.603	0.610	0.614	0.642
Joe Ciccone	0.508	0.570	0.560	0.601	0.595	0.599	0.608	0.635
Lonnie Waliczek	0.539	0.568	0.577	0.598	0.601	0.619	0.618	0.631
Michael Haugen Jr.	0.547	0.561	0.570	0.606	0.602	0.608	0.611	0.649
Michael Machuga	0.527	0.558	0.573	0.593	0.607	0.629	0.629	0.660
Mika Koivuniemi	0.540	0.588	0.581	0.616	0.601	0.617	0.631	0.648
Mike DeVaney	0.522	0.561	0.587	0.591	0.594	0.604	0.617	0.638
Mike Edwards	0.556	0.576	0.581	0.613	0.599	0.616	0.622	0.650
Mike Fagan	0.554	0.571	0.564	0.610	0.620	0.607	0.622	0.629
Mike Scroggins	0.553	0.603	0.610	0.613	0.612	0.624	0.612	0.652
Mike Wolfe	0.563	0.585	0.624	0.627	0.614	0.620	0.617	0.636
Norm Duke	0.533	0.610	0.577	0.604	0.611	0.617	0.618	0.654
Parker Bohn III	0.572	0.609	0.606	0.623	0.621	0.625	0.630	0.659
Patrick Healey JR	0.561	0.549	0.597	0.599	0.605	0.614	0.618	0.631
Patrick Allen	0.581	0.604	0.592	0.622	0.611	0.626	0.616	0.648
Paul Fleming	0.530	0.534	0.575	0.596	0.605	0.614	0.615	0.624
Pete Weber	0.554	0.583	0.586	0.608	0.601	0.612	0.614	0.641
Rick Lawrence	0.552	0.556	0.577	0.592	0.609	0.604	0.625	0.622
Rick Steelsmith	0.549	0.580	0.613	0.591	0.609	0.607	0.609	0.634
Robert Smith	0.548	0.554	0.582	0.613	0.591	0.612	0.616	0.657
Ryan Shafer	0.541	0.559	0.610	0.588	0.603	0.611	0.612	0.646
Sean Rash	0.557	0.576	0.584	0.598	0.600	0.612	0.614	0.639
Steve Jaros	0.542	0.603	0.617	0.595	0.603	0.614	0.615	0.625
Steve Wilson	0.550	0.600	0.600	0.613	0.598	0.616	0.621	0.617
Tim Criss	0.552	0.604	0.564	0.580	0.587	0.617	0.624	0.597
Tom Baker	0.541	0.571	0.575	0.594	0.586	0.600	0.618	0.614
Tommy Delutz Jr.	0.552	0.586	0.573	0.612	0.606	0.598	0.613	0.647
Tommy Jones	0.563	0.592	0.589	0.604	0.597	0.615	0.608	0.651
Tony Reyes	0.532	0.581	0.592	0.608	0.581	0.600	0.615	0.639
Walter Ray Williams Jr.	0.556	0.594	0.619	0.626	0.620	0.631	0.631	0.663
Wes Malott	0.540	0.592	0.593	0.603	0.612	0.626	0.620	0.653

Table 6:

Posterior means for T_k’s with no Bayesian shrinkage, i.e. the empirical tranisition probabilities.

	−−−−	−−−	−−	−	X	XX	XXX	XXXX
Amleto Monacelli	0.600	0.548	0.555	0.560	0.571	0.576	0.602	0.607
Bill O’Neill	0.533	0.696	0.595	0.623	0.603	0.625	0.636	0.628
Bob Learn Jr.	0.514	0.600	0.585	0.613	0.618	0.587	0.618	0.609
Brad Angelo	0.635	0.636	0.613	0.584	0.600	0.607	0.615	0.629
Brian Himmler	0.472	0.548	0.593	0.599	0.595	0.595	0.643	0.642
Brian LeClair	0.326	0.738	0.659	0.585	0.535	0.537	0.560	0.551
Brian Voss	0.494	0.602	0.647	0.636	0.629	0.608	0.638	0.671
Brian Kretzer	0.581	0.538	0.541	0.548	0.578	0.544	0.572	0.578
Chris Barnes	0.610	0.614	0.558	0.634	0.640	0.660	0.642	0.668
Chris Collins	0.645	0.519	0.527	0.602	0.590	0.566	0.573	0.707
Chris Johnson	0.541	0.400	0.512	0.611	0.588	0.598	0.641	0.576
Chris Loschetter	0.367	0.628	0.565	0.580	0.573	0.646	0.614	0.632
Danny Wiseman	0.505	0.647	0.597	0.607	0.613	0.629	0.620	0.624
Dave D’Entremont	0.552	0.496	0.480	0.564	0.556	0.545	0.621	0.598
David Traber	0.586	0.581	0.724	0.644	0.619	0.581	0.671	0.652
Dick Allen	0.557	0.530	0.573	0.580	0.577	0.640	0.604	0.703
Doug Kent	0.540	0.586	0.542	0.606	0.559	0.634	0.559	0.646
Eugene McCune	0.610	0.609	0.564	0.602	0.569	0.638	0.650	0.658
Hugh Miller	0.551	0.565	0.522	0.612	0.582	0.566	0.621	0.594
Jack Jurek	0.574	0.568	0.594	0.595	0.616	0.564	0.556	0.630
Jason Couch	0.600	0.544	0.677	0.639	0.626	0.609	0.644	0.652
Jason Hurd	0.466	0.645	0.645	0.564	0.597	0.581	0.574	0.622
Jeff Carter	0.500	0.549	0.550	0.614	0.604	0.605	0.591	0.645
Joe Ciccone	0.424	0.552	0.536	0.598	0.588	0.575	0.567	0.626
Lonnie Waliczek	0.514	0.532	0.562	0.590	0.600	0.637	0.619	0.611
Michael Haugen Jr.	0.548	0.506	0.549	0.609	0.603	0.598	0.573	0.663
Michael Machuga	0.488	0.523	0.562	0.585	0.612	0.651	0.668	0.675
Mika Koivuniemi	0.530	0.599	0.577	0.623	0.601	0.622	0.673	0.654
Mike DeVaney	0.475	0.531	0.586	0.582	0.586	0.592	0.613	0.634
Mike Edwards	0.589	0.565	0.575	0.622	0.596	0.623	0.647	0.662
Mike Fagan	0.573	0.556	0.549	0.615	0.634	0.600	0.637	0.617
Mike Scroggins	0.567	0.639	0.623	0.618	0.618	0.636	0.598	0.661
Mike Wolfe	0.623	0.596	0.657	0.648	0.627	0.632	0.616	0.630
Norm Duke	0.500	0.652	0.570	0.603	0.618	0.622	0.619	0.665
Parker Bohn III	0.644	0.659	0.618	0.635	0.636	0.641	0.668	0.671
Patrick Healey JR	0.598	0.496	0.603	0.595	0.608	0.616	0.618	0.620
Patrick Allen	0.661	0.635	0.594	0.631	0.616	0.639	0.611	0.652
Paul Fleming	0.484	0.407	0.558	0.584	0.610	0.619	0.596	0.586
Pete Weber	0.569	0.587	0.584	0.610	0.599	0.612	0.601	0.641
Rick Lawrence	0.574	0.478	0.562	0.574	0.621	0.581	0.676	0.575
Rick Steelsmith	0.564	0.579	0.639	0.575	0.618	0.596	0.560	0.621
Robert Smith	0.552	0.500	0.576	0.622	0.579	0.611	0.607	0.676
Ryan Shafer	0.533	0.532	0.620	0.580	0.603	0.610	0.597	0.650
Sean Rash	0.619	0.559	0.576	0.588	0.596	0.609	0.583	0.634
Steve Jaros	0.528	0.639	0.634	0.589	0.604	0.616	0.605	0.608
Steve Wilson	0.581	0.698	0.624	0.629	0.588	0.628	0.647	0.545
Tim Criss	0.568	0.652	0.545	0.555	0.570	0.626	0.653	0.531
Tom Baker	0.529	0.555	0.565	0.586	0.568	0.579	0.617	0.580
Tommy Delutz Jr.	0.571	0.596	0.558	0.619	0.609	0.573	0.588	0.656
Tommy Jones	0.590	0.606	0.589	0.604	0.594	0.617	0.584	0.658
Tony Reyes	0.477	0.584	0.596	0.614	0.548	0.567	0.594	0.636
Walter Ray Williams Jr.	0.578	0.616	0.636	0.637	0.632	0.650	0.664	0.677
Wes Malott	0.526	0.610	0.596	0.602	0.619	0.643	0.628	0.663

Table 7:

Shrunken estimates of most probable (across all players) non-strike frame breakdown probabilities (columns 1–5) and probability of spare (column 6).

	(9,1)	(8,2)	(7,3)	(8,1)	(7,2)	Any spare
Amleto Monacelli	0.537	0.142	0.060	0.065	0.057	0.774
Bill O’Neill	0.561	0.125	0.064	0.058	0.050	0.779
Bob Learn Jr.	0.595	0.124	0.052	0.053	0.048	0.810
Brad Angelo	0.559	0.112	0.071	0.058	0.048	0.778
Brian Himmler	0.580	0.111	0.063	0.060	0.052	0.786
Brian LeClair	0.534	0.129	0.067	0.062	0.051	0.769
Brian Voss	0.585	0.120	0.064	0.052	0.051	0.795
Brian Kretzer	0.552	0.126	0.065	0.066	0.053	0.781
Chris Barnes	0.624	0.124	0.039	0.059	0.048	0.810
Chris Collins	0.544	0.124	0.064	0.066	0.045	0.768
Chris Johnson	0.532	0.114	0.084	0.060	0.055	0.763
Chris Loschetter	0.542	0.135	0.074	0.069	0.052	0.782
Danny Wiseman	0.611	0.104	0.056	0.058	0.049	0.804
Dave D’Entremont	0.521	0.119	0.076	0.061	0.059	0.753
David Traber	0.646	0.111	0.045	0.058	0.032	0.828
Dick Allen	0.538	0.133	0.070	0.058	0.052	0.777
Doug Kent	0.597	0.117	0.057	0.060	0.043	0.801
Eugene McCune	0.560	0.120	0.069	0.060	0.044	0.789
Hugh Miller	0.576	0.143	0.056	0.059	0.059	0.798
Jack Jurek	0.566	0.112	0.072	0.063	0.054	0.777
Jason Couch	0.605	0.115	0.059	0.055	0.052	0.808
Jason Hurd	0.538	0.121	0.074	0.064	0.049	0.763
Jeff Carter	0.531	0.126	0.068	0.067	0.051	0.763
Joe Ciccone	0.556	0.129	0.054	0.057	0.051	0.780
Lonnie Waliczek	0.569	0.113	0.058	0.065	0.049	0.775
Michael Haugen Jr.	0.618	0.116	0.055	0.049	0.049	0.816
Michael Machuga	0.554	0.124	0.065	0.061	0.048	0.777
Mika Koivuniemi	0.594	0.107	0.068	0.056	0.053	0.800
Mike DeVaney	0.555	0.121	0.068	0.066	0.048	0.785
Mike Edwards	0.575	0.120	0.068	0.064	0.050	0.795
Mike Fagan	0.555	0.122	0.056	0.061	0.045	0.775
Mike Scroggins	0.623	0.122	0.049	0.052	0.049	0.818
Mike Wolfe	0.591	0.118	0.059	0.060	0.046	0.796
Norm Duke	0.653	0.108	0.048	0.055	0.041	0.835
Parker Bohn III	0.618	0.101	0.051	0.064	0.050	0.792
Patrick Healey JR	0.584	0.108	0.057	0.059	0.049	0.779
Patrick Allen	0.619	0.111	0.054	0.053	0.049	0.815
Paul Fleming	0.600	0.132	0.052	0.053	0.047	0.816
Pete Weber	0.570	0.119	0.077	0.056	0.048	0.800
Rick Lawrence	0.600	0.124	0.059	0.058	0.039	0.812
Rick Steelsmith	0.585	0.114	0.057	0.061	0.047	0.788
Robert Smith	0.570	0.124	0.068	0.059	0.044	0.789
Ryan Shafer	0.587	0.121	0.059	0.063	0.056	0.794
Sean Rash	0.543	0.115	0.078	0.062	0.056	0.766
Steve Jaros	0.602	0.123	0.054	0.057	0.055	0.811
Steve Wilson	0.587	0.126	0.055	0.059	0.048	0.799
Tim Criss	0.592	0.123	0.055	0.062	0.055	0.795
Tom Baker	0.600	0.114	0.061	0.059	0.050	0.810
Tommy Delutz Jr.	0.595	0.115	0.061	0.059	0.051	0.801
Tommy Jones	0.549	0.129	0.065	0.066	0.059	0.772
Tony Reyes	0.576	0.121	0.062	0.061	0.052	0.790
Walter Ray Williams Jr.	0.691	0.100	0.044	0.048	0.040	0.852
Wes Malott	0.582	0.112	0.059	0.059	0.048	0.784

Table 8:

Unshrunken (i.e. empirical) estimates of most probable (across all players) non-strike frame breakdown probabilities (columns 1–5) and probability of spare (column 6).

	(9,1)	(8,2)	(7,3)	(8,1)	(7,2)	Any spare
Amleto Monacelli	0.519	0.149	0.060	0.066	0.060	0.767
Bill O’Neill	0.549	0.126	0.066	0.057	0.050	0.771
Bob Learn Jr.	0.601	0.126	0.048	0.049	0.047	0.818
Brad Angelo	0.554	0.110	0.072	0.058	0.048	0.773
Brian Himmler	0.578	0.108	0.064	0.061	0.052	0.782
Brian LeClair	0.506	0.133	0.071	0.064	0.051	0.754
Brian Voss	0.585	0.119	0.066	0.049	0.052	0.796
Brian Kretzer	0.543	0.127	0.066	0.067	0.054	0.775
Chris Barnes	0.632	0.125	0.035	0.059	0.047	0.813
Chris Collins	0.518	0.126	0.066	0.071	0.042	0.750
Chris Johnson	0.503	0.110	0.096	0.060	0.057	0.744
Chris Loschetter	0.526	0.140	0.079	0.072	0.052	0.777
Danny Wiseman	0.617	0.099	0.055	0.057	0.048	0.805
Dave D’Entremont	0.496	0.117	0.080	0.061	0.061	0.734
David Traber	0.677	0.106	0.038	0.057	0.023	0.844
Dick Allen	0.514	0.138	0.075	0.056	0.053	0.767
Doug Kent	0.599	0.116	0.056	0.061	0.042	0.802
Eugene McCune	0.553	0.120	0.072	0.060	0.042	0.786
Hugh Miller	0.572	0.155	0.054	0.058	0.064	0.799
Jack Jurek	0.559	0.108	0.077	0.064	0.056	0.770
Jason Couch	0.609	0.113	0.058	0.053	0.052	0.809
Jason Hurd	0.516	0.120	0.079	0.066	0.048	0.748
Jeff Carter	0.506	0.129	0.072	0.070	0.052	0.747
Joe Ciccone	0.548	0.131	0.052	0.056	0.052	0.775
Lonnie Waliczek	0.561	0.109	0.057	0.067	0.049	0.765
Michael Haugen Jr.	0.633	0.114	0.053	0.045	0.048	0.825
Michael Machuga	0.546	0.124	0.066	0.061	0.047	0.772
Mika Koivuniemi	0.595	0.104	0.069	0.055	0.054	0.800
Mike DeVaney	0.548	0.121	0.069	0.067	0.048	0.782
Mike Edwards	0.572	0.120	0.071	0.065	0.050	0.794
Mike Fagan	0.548	0.122	0.055	0.061	0.044	0.768
Mike Scroggins	0.630	0.122	0.046	0.050	0.049	0.823
Mike Wolfe	0.592	0.116	0.059	0.060	0.045	0.795
Norm Duke	0.666	0.105	0.046	0.054	0.040	0.843
Parker Bohn III	0.624	0.096	0.049	0.064	0.050	0.790
Patrick Healey JR	0.584	0.105	0.056	0.059	0.049	0.774
Patrick Allen	0.625	0.109	0.053	0.052	0.049	0.819
Paul Fleming	0.607	0.136	0.048	0.051	0.047	0.824
Pete Weber	0.566	0.118	0.080	0.055	0.047	0.799
Rick Lawrence	0.607	0.126	0.059	0.057	0.034	0.819
Rick Steelsmith	0.586	0.111	0.055	0.061	0.046	0.785
Robert Smith	0.566	0.125	0.070	0.059	0.042	0.788
Ryan Shafer	0.586	0.121	0.058	0.063	0.057	0.793
Sean Rash	0.519	0.112	0.087	0.064	0.059	0.748
Steve Jaros	0.607	0.123	0.052	0.056	0.056	0.815
Steve Wilson	0.589	0.130	0.051	0.059	0.046	0.802
Tim Criss	0.594	0.124	0.053	0.062	0.056	0.796
Tom Baker	0.605	0.113	0.062	0.059	0.051	0.813
Tommy Delutz Jr.	0.599	0.113	0.061	0.059	0.052	0.803
Tommy Jones	0.542	0.130	0.066	0.066	0.061	0.767
Tony Reyes	0.573	0.121	0.063	0.062	0.053	0.788
Walter Ray Williams Jr.	0.711	0.096	0.041	0.046	0.038	0.863
Wes Malott	0.579	0.110	0.058	0.058	0.048	0.779

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Published Online: 2018-09-05

Published in Print: 2018-11-27

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Articles in the same Issue

https://doi.org/10.1515/jqas-2017-0081

Keywords for this article

Bayesian hierarchical model; bowling; hot hand; Markov model