USING PRIORS TO IMPROVE ESTIMATES OF MUSIC STRUCTURE
Jordan B. L. Smith
Masataka Goto
National Institute of Advanced Industrial Science and Technology (AIST), Japanjordan.smith@aist.go.jp, m.goto@aist.go.jp

ABSTRACT
Existing collections of annotations of musical structure
possess many strong regularities: for example, the lengthsof segments are approximately log-normally distributed, as
is the number of segments per annotation; and the lengthsof two adjacent segments are highly likely to have an integer ratio. Since many aspects of structural annotations arehighly regular, but few of these regularities are taken into
account by current algorithms, we propose several methods of improving predictions of musical structure by using
their likelihood according to prior distributions. We test theuse of priors to improve a committee of basic segmentation
algorithms, and to improve a committee of cutting-edge
approaches submitted to MIREX. In both cases, we areunable to improve on the best committee member, meaning that our proposed approach is outperformed by simple parameter tuning. The same negative result was founddespite incorporating the priors in multiple ways. To explain the result, we show that although there is a correlation overall between output accuracy and prior likelihood,
the weakness of the correlation in the high-likelihood region makes the proposed method infeasible. We suggestthat to improve on the state of the art using prior likelihoods, these ought to be incorporated at a deeper level ofthe algorithm.

Figure 1. Proposed system overview.
This is important to recognize, since structure is ambiguous: for any piece, there are often multiple ways ofinterpreting it. As a result, we might never expect a purely
bottom-up approach to be 100% correct; we need to alsomodel the top-down influence, or what the listener ‘brings’
to the analysis.
For instance, consider the following analysis of a pieceof music, with the A sections each 10 seconds long, and B
200 seconds long:[-A-|-A-|-A-|------------B------------]

1. INTRODUCTION
One reason that the perception of structure in music is sucha complex and compelling phenomenon is that it is a combination of ‘bottom-up’ and ‘top-down’ processes. It isbottom-up in the sense that a listener first performs grouping on short timescales before understanding the groupingat large timescales, but it is top-down in the sense that one
has global expectations that can affect the way one perceives the music. For example, when hearing a new pop
song for the first time, we expect there to be a chorus; evenon our first hearing, we may identify the chorus partway
through a song and already expect it to repeat later. Afterhearing a verse and a chorus, each 32 beats long, we may
expect the bridge to be the same length when it starts.
c Jordan B. L. Smith, Masataka Goto. Licensed under aCreative Commons Attribution 4.0 International License (CC BY 4.0).
Attribution: Jordan B. L. Smith, Masataka Goto. “Using Priors To
Improve Estimates of Music Structure”, 17th International Society forMusic Information Retrieval Conference, 2016.

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Even without knowing the piece of music, we can tell thisis an unlikely analysis; it seems wrong to have the segments of the piece sized so asymmetrically. This example
hints that the space of plausible analyses is limited (evenif it is huge), and that listeners’ intuitions about these limits inform the annotation process. Is there a way to embedsuch intuitions into music structure analysis algorithms?
Can we employ a kind of ‘top-down’ critic to assess thelikelihood of a given analysis?
We propose a system to accomplish this, illustrated in
Figure 1. The inputs to the system are: (a) a song to analyze, and (b) a set of probability density functions (PDFs)
estimated from a corpus of annotations. The input songis analyzed by a set of algorithms (step 1); the prior likelihood of each output is computed (step 2); and the estimateddescription with the highest prior likelihood is chosen (step3). In contrast to the usual parameter tuning approach, inwhich a single parameter setting is fixed after evaluating
performance over a corpus of songs, in our approach parameters can be tuned for each song, on the basis of priorlikelihood.

Proceedings of the 17th ISMIR Conference, New York City, USA, August 7-11, 2016
1.1 Related work
Most algorithms do at least some domain knowledge-basedtuning, by putting a lower and/or upper bound on the lengthof segments, or by filtering features to reduce variations atcertain timescales. These are important steps because, although musical structure is hierarchical, algorithms rarely
attempt to predict this hierarchy and are evaluated only at asingle level. (This status quo has been challenged by [5].)However, a few algorithms have made greater use of
domain knowledge, and their success has been noteworthy. Among the first optimization-based approaches tostructure analyis was [7], who explicitly sought to define(in a top-down way) what constitutes a “good” analysis(i.e., one more likely to be in the space of plausible solutions). Later, [10] estimated the median segment lengthof a piece and used this as the preferred segment length inits search for an optimal segmentation; at the time, theiralgorithm outperformed the leader at MIREX. The AutoMashUpper system also uses a cost-based approach, rewarding solutions with “good” segment durations of 2, 4,
or 8 downbeats, and penalizing ones deemed less likely,
like 7 or 9 [2]. In the symbolic domain, [9] also used acost-minimization approach, with costs increased for segments of unlikely duration or unlikely melodic contour; on
one dataset, the approach outperformed a pack of leading
melodic-segmentation algorithms.
The most direct way to use domain knowledge is touse supervised learning. Two examples include [14, 15],
who each used machine learning to classify short excerpts
as boundaries or not based on their resemblance to othershort excerpts known to be boundaries. The performanceof [15] exceeded the best MIREX result by nearly 10%
f -measure for both 0.5- and 3-second thresholds, an enormous achievement.
Our intuition about what constrains the space of plausible analyses, as well as the success of previous algorithms
in using domain knowledge and priors learned from corpora, suggest that taking full advantage of this prior knowledge is essential to designing effective algorithms.
In the next section, we survey some of these regularities,
and explore the extent to which prior algorithms adhere tothem. We detail our proposed algorithms and report ourexperimental results in Section 3. Alas, despite the solid
foundations, no approach will be found to work. The significance of this negative result, and possible explanations
for it, are discussed in Section 4.
2. REGULARITIESIn this section we briefly survey some regularities found in
the SALAMI corpus of annotations [12], and describe therelationship between these regularities and algorithms thathave participated in MIREX campaigns from 2012–14.
Although the time scale of the SALAMI annotations
was not explicitly constrained in the Annotator’s Guide 1 ,the length of annotated segments in the SALAMI corpus
1 Available at http://ddmal.music.mcgill.ca/
research/salami/annotations.

555(a) log segment length(b) log number of segments(c) log ratio to median segment length

Figure 2. Estimated PDFs of three properties for annotations in SALAMI corpus; x-axis gives the value of theproperty, y-axis gives its relative probability. PDFs fromlarge-scale segments shown in black, from small-scale segments in gray. In (c), the vertical axis is clipped to show
detail; the gray line extends upwards to just over 0.1.

is roughly log-normally distributed, for both hierarchicallevels. Figure 2a shows the PDF of the log segment length(P (log Li )) for the large and small hierarchical levels in
SALAMI; this and all other PDFs in this paper were foundusing kernel density estimation (KDE). The number of segments within a piece (N ) is also log-normally distributed(Figure 2b). If we take the log ratio of each segment’s
length to the median length of segments within that piece(log(Li /Lmed )), we obtain a PDF strongly concentrated atlog (1) = 0 (see Figure 2c), with additional spikes near
±0.693, or log (2) and log (1/2), for segments of twiceor half the median length. There is even more detail if
we look at the log length ratio between adjacent segments(log(Li /Li+1 )), a histogram of which is shown in Figure 3. Note that all the prominent peaks occur at ratiosof small numbers. This makes sense if we consider thatsegments are usually a whole number of measures long.
These properties are not specific to SALAMI annotations;similar distributions were reported by [1] for a completely
different corpus of annotations.
How closely do algorithms model these properties of
the annotations? We looked at three years of participantsin the MIREX Structural Segmentation task, 2012–14, andestimated PDFs for the same properties. Some examples
are shown in Figure 4. Figure 4a shows PDFs for segment
length estimated from each algorithm individually: somehew closely to the ground truth, but the majority underestimate the mean segment length. (Since precision is harderto achieve than recall, oversegmentation usually leads to556

Proceedings of the 17th ISMIR Conference, New York City, USA, August 7-11, 2016(a) PDFs of segment duration (in seconds)

Figure 3. Histogram of log(Li /Li+1 ) estimated fromSALAMI annotations (y-axis truncated at 1/3 maximum).
As shown, nearly every spike represents an integer ratio ofsegment lengths.
better evaluation scores. [13])If segment length seems like a weak prior, consider
instead Figure 4b, which compares PDFs for the log ratio of adjacent segment length. The characteristic sidelobes representing the high frequency of half- and doublelength segments are prominent in only two of the algorithms, RBH1 and RBH3 (2013). This is likely because
the algorithm [8] expects boundaries to occur on an 8measure metrical grid, and snaps estimated boundaries tothis grid. Performance (evaluated with f -measure and3-second threshold) was mixed: RBH1 was close to thestate of the art in 2013, while RBH3 was below-average. 2
On the other hand, the next-strongest side-lobes belong to
SUG1, the second-best algorithm overall. SUG1 uses a
convolutional neural network to classify short excerpts as
containing boundaries or not; the method ends by picking peaks from a boundary-likelihood curve, without postprocessing [15]. Although SUG1 obviously learns fromannotated data, it learns from low-level features (a mel
spectrogram) rather than high-level attributes like segment
length ratios.
Does the fitness of the algorithms to the SALAMIderived priors actually have an impact on their performance? We found this to be true by looking at the correlation between algorithm performance and prior likelihood.
We took the output of the 18 unique segmentation algorithms that participated in MIREX from 2012–14 3 , andfor each algorithm, computed the average log-likelihoodof its estimated segments based on the KDE-derived PDFs
from SALAMI. We also took the average performance of
the algorithms on the three boundary retrieval metrics (f measure, precision, and recall) with a threshold of 3 seconds. Figure 5 shows the correlation between the mean
log-likelihoods (of various segment properties) and theevaluation metrics. There is a weak to moderate correla2 Evaluation results in this paper differ from those reported at MIREX,
since we re-evaluated the algorithm output with a 5-second ‘warm-up’
applied: boundaries within the first and final 5 seconds of pieces wereignored. This leads to lower results overall but better differentiation between algorithms.
3 Of the 24 participants in these years, 5 used the same segmentation
algorithm as another, and the data for one (FK2) were posted later thanthe others, and were excluded.

(b) PDFs of log ratio of length of adjacent segments

Figure 4. PDFs of properties estimated from SALAMI
annotations (black dotted line) and from the output of
MIREX algorithms (each algorithm in a different colour).
tion between likelihood and f -measure for each of theseproperties, usually attributable to a strong correlation between likelihood and either precision or recall.
We have seen that most algorithms deviate substantiallyfrom the corpus; the algorithms’ descriptions simply don’t‘look’ like the ground truth. Also, there is some evidence
that an algorithm’s accuracy is related to the prior likelihood of its output. Hence, it seems reasonable to ask: canwe improve on these algorithms, or any set of algorithms,
by maximizing their fitness to the priors?
3. USING PRIORS TO IMPROVE A COMMITTEE
Our proposed system is simple: for a single audio file,(1) run several existing structural analysis algorithms, (2)compute the log-likelihood of each prediction with respect
to a corpus, and (3) choose the output that maximizes this.
(See Figure 1.) For each of these steps, there are many
ways to proceed.
3.1 Assembling a committee
We assembled two committees of algorithms: a committee of multiple parameterizations of two basic approaches(Foote [3] and Serra et al. [11]), and a committee of approaches drawn from MIREX. In the first case, we testProceedings of the 17th ISMIR Conference, New York City, USA, August 7-11, 2016

557

it, along with the reported algorithm performance, for a
MIREX committee of 23 members. We restricted ourselves to the SALAMI portion of the MIREX evaluation, which overlaps significantly with the public half of
SALAMI but is not identical.
3.2 Computing likelihoods
We looked at the distribution of several attributes of the
SALAMI corpus, listed below. Of these, A1−4 are estimated on a per-segment basis and A5−9 are global attributes of a description.
• A1 Segment length (Li )• A2 Fractional segment length (Li / song length)• A3 Ratio of Li to median segment length• A4 Ratio of adjacent segment lengths (Li /Li+1 )• A5 Median segment length (median of Li )• A6 Number of segments• A7 Minimum segment length• A8 Maximum segment length• A9 Standard deviation of segment length
For attributes A1−4 , we took the average across segments. Although log likelihoods are designed to besummed, taking the sum of log P (Li ) would punish descriptions with more boundaries, regardless of how probable the segments are. (In fact, we did test taking the sum
instead of the mean across segments, and the results weregenerally much poorer.)
Figure 5. Scatter plots of mean log likelihood of algorithm output (x-axis) and f -measure, precision and recall(y-axis), for 18 MIREX segmentation algorithms, 2012–
14. Correlations (Pearson’s R) and lines of best fit are
shown.
whether a set of more strictly bottom-up segmentation approaches can be improved with the top-down likelihoodmaximizing process; in the second case, we test whether aset of already-optimized algorithms can also be improved.
Foote uses a checkerboard kernel to identify discontinuities between homogeneous regions in a self-similarity matrix (SSM), and his remains a classic approach to segmentation. Although surpassed in evaluations such as MIREX,
the simplicity and effectiveness of the algorithm means it
is still commonly used as a model to improve upon (e.g.,see [4]). In contrast to Foote, Serra et al. [11] aim to use
both repetition and novelty to locate boundaries. In practice, both algorithms require several design choices: whichaudio features to use, what amount of smoothing to apply,
etc. We ran each algorithm with a small factorial rangeof settings, including three features (HPCP, MFCCs, andtempograms), for a total of 40 unique settings—hence, 40committee members. We ran the algorithms on 773 songs
within the public SALAMI corpus (version 1.9). Feature extraction and algorithm computation were both easily
handled using MSAF [6].
The output of the algorithms that participated in
MIREX is publically available, so we simply assembled

3.3 Electing a winner
Once we have computed all of the log likelihoods, howshould they be combined, and how should we use thesevalues to elect an answer? Without any a priori reason toprefer one over another, we tested multiple approaches:• choose the description that maximizes the likelihoodof attribute Ai ;• choose the description that maximizes a summary
statistic over all attributes;• use a linear model to predict f -measure based on thelikelihoods;• use a linear model with interactions;• use a quadratic model.
As two summary statistics, we used the sum and theminimum of the log likelihoods of Ai . Using the sum optimizes the general fitness; using the minimum penalizes
descriptions with any unlikely attributes.
3.4 Experiment and results
With 5-fold cross-validation, we tested all versions of thealgorithm, using both the Foote/Serra and MIREX committees. For each fold, the prior PDFs were estimated only
using annotations from the training set.
As a baseline, we used simple parameter tuning: i.e.,
simply pick the committee member with the greatest average success on the training set. For reference, we also computed the mean f -measure of all committee members, and558

Proceedings of the 17th ISMIR Conference, New York City, USA, August 7-11, 2016
Attribute
A1 (mean)A2 (mean)A3 (mean)A4 (mean)
A5
A6
A7
A8
A
P9
Ai
Min AiLinear modelInteractions model
Quadratic model
Baseline
Committee mean
Theoretical max

f (3 sec)
0.42300.4156
0.4176
0.41940.35970.37810.06030.39070.3956
0.42600.42060.43990.44510.44940.44390.28260.6015f (0.5 sec)
0.1051
0.09580.11400.10720.08630.09910.0124
0.0961
0.09500.10930.10460.08450.06880.07390.1151
0.06910.2572

Table 1. Average f -measure (at two thresholds) achievedby different decision criteria for Foote-and-Serra committee.
the theoretical maximum—i.e., the average of the highestscoring estimates for each song.
The results are shown in Tables 1 and 2. Among all
the variations of the proposed method, there were only two
instances that surpassed the baseline: the quadratic and interactions models for the Foote-Serra committee, with a
3-second tolerance level. They surpassed the baseline f measure by 0.0055 and 0.0012, respectively. Given thenumber of trials conducted, this small amount of successcould easily have come by chance.
4. DISCUSSION
Negative results are not normally conclusive: in this case,the reader may suspect that with a small twist, our proposed method may yet succeed. For example, what if
we examined subsets by genre, or considered conditional
probabilities? In fact, this process of tweaking is how ourexperiments came about. Our first effort to solve the problem used a small committee of solely Foote-based algorithms, and a set of four log likelihoods. When tests withthis initial system gave us a negative result, we tried varying each of the parts of the system—adding more members to the committee, including more PDFs, using increasingly sophisticated regression approaches—until we
had assembled the large-scale experiment reported here.
And we conducted several more informal tests—looking
at subsets of the data, varying the method of characterizing
the PDFs (instead of with KDE, they can be modelled withplain histograms, or normal curves can be fitted to some
distributions), looking at subcommittees (e.g., removing
top-performing and low-performing outlier members) andcomputing two-dimensional priors (to model, for example,
the fact that segment length is not independent of when asegment begins)—all to no avail.

Attribute
A1 (mean)A2 (mean)A3 (mean)A4 (mean)
A5
A6
A7
A8
A
P9
Ai
Min AiLinear modelInteractions model
Quadratic model
Baseline
Committee mean
Theoretical max

f (3 sec)
0.62730.34870.34870.34870.3916
0.37680.3487
0.46620.42330.62730.62730.55910.62730.62730.62730.28260.7345f (0.5 sec)
0.27330.0996
0.0996
0.09960.13850.15940.09960.13560.15140.27330.27330.40050.40050.40050.40050.06910.5157

Table 2. Average f -measure (at two thresholds) achievedby different decision criteria for MIREX committee.
The consistency of the negative result—only two trials
out of 56 exceeded the baseline, and only by the slimmestof margins—suggests a dead end. But in order to draw
conclusions from this negative result, we must try to understand why the approach failed.
Earlier, in Figure 5, we saw that algorithm performance
could, over many trials, correlate with the prior likelihoodof their output. But what happens when we dig deeper
and look at the relationship between each individual output’s correctness and its likelihood, as in Figure 6? On the
one hand, there is a clear positive trend overall, since there
are no examples in the upper-left corner—that is, there are
no predictions that have low likelihood but that are close
to correct. And the examples with the highest f -measure
are also among those with the highest likelihoods. Thanks
to this relationship, the committees can, despite the noise,
generally choose an output that is at least, or slightly better than, the committee’s average; that’s why, in Tables 1
and 2, nearly all of the algorithms exceeded the average
result of the committee.
On the other hand, trying to find the high-f -measurepredictions based on their prior likelihood is clearly futile
when we consider only the rightmost region of the plot, azoomed-in portion of which is shown in the lower part of
Figure 6. Even these predictions, with the greatest fit to thepriors, range widely in accuracy: there are plenty above thebaseline (0.44), but also plenty below it, including a large
number of predictions that contain zero correct boundaries.
Figure 6 shows that having a high log-likelihood is a necessary but not sufficient condition to be correct, and it is acondition that most algorithms already achieve.
The uppermost points in Figure 6 represent a few lucky,
perfect estimates of the true structure. Their distribution
reveals another important point: that although the prior
PDFs derive from the ground truth, the prior likelihood ofProceedings of the 17th ISMIR Conference, New York City, USA, August 7-11, 2016

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Figure 7. Two algorithmic estimates, compared to theground truth (middle). The estimates differ somewhat in
the likelihood of A4 (adjacent segment length), but drastically in f -measure. The song is “Rock With You” by
Michael Jackson, SALAMI ID 1616.

3. Among high-likelihood descriptions, the correlation
between success and likelihood is much weaker:
many likely descriptions are poor, and many annotations have low likelihood.
5. CONCLUSION AND FUTURE WORKP
Figure 6. Above: scatter plot oflog P (Ai ) (x-axis)
vs. f -measure (y-axis, 3-second threshold) of algorithm
outputs for Foote/Serra committee on all data. (The heavy
horizontal lines are caused by the fact that f -measure isoften the product of common fractions.) Below: inset ofplot indicated by rectangle.
many annotations is moderate. The fat tails of the PDFs in
Figure 2 represent a large set of descriptions that are unlikely to ever be predicted by a prior likelihood-based approach. For example, consider the analyses shown in Figure 7. 4 One algorithm achieved a perfect f -measure (with
3-second threshold), and the likelihood of the description(measured with respect to attribute A4 ) was close to thatof the annotated description. But a second estimate had aslightly higher prior likelihood thanks to its more consistent segment lengths, and a very poor f -measure.
To sum up the factors that appear to limit the effectiveness of our approach:
1. Although a high f -measure tends to come witha higher prior likelihood, the reverse is not true:
plenty of highly probable descriptions are very poor.
2. The moderate correlation between algorithm success and prior likelihood is irrelevant, since weare interested only in the high-likelihood region of
estimated descriptions.
4 Although the MIREX data are anonymized, many songs can be identified by comparing the ground truth to known datasets. [13]We proposed and tested a novel committee-based approachto structural analysis. We motivated the approach by discussing the strong regularities displayed by annotations ofmusic structure. But after a long stretch of negative results,
we have concluded that the approach seems unviable: therelationship between a description’s prior likelihood andits evaluated score seems to be too weak, especially in thehigh-likelihood region we are interested in.
We began the article by pointing out some mismatches
between the properties of algorithmic estimates of structure and the ground truth, and we suggested that thismay be because algorithms do not model top-down factors in perception. For a listener, top-down factors interact
with bottom-up factors; in contrast, our algorithm applies
bottom-up considerations first (by collecting the committee of estimates), and then applies the top-down considerations post hoc. This may be the central weakness of our algorithm. Perhaps, if the top-down influence were modelled
earlier on, an estimate like the top one in Figure 7 could be
fine-tuned, the boundaries shifted slightly to give a moreprobable output, rather than rejected early on because of
its low likelihood. One algorithm that is ready to test thisas future work is the optimization approach of [10]. Although the authors model only a few basic priors, it could
be improved by including more.
6. ACKNOWLEDGEMENTS
A version of this work was shown to participants at the
Dagstuhl Workshop on Computational Music Structure
Analysis. We are very grateful to the attendees for theirhelpful suggestions, and to the workshop organizers andthe Leibniz Center for Informatics for hosting us. This
work was supported in part by OngaCREST, CREST, JST.

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