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Bayesian Hierarchical Models for Tailoring Metrics Thresholds Neil Ernst University of Victoria Software Engineering @neilernst

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Bayesian Hierarchical Models for X Neil Ernst University of Victoria Software Engineering @neilernst

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Bottom line (SE) We can use hierarchical models to tailor metrics to specific contexts → In this case, with 50% drop in RMSE (Method) Hierarchical modeling with Bayesian inference fit SE data very well

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Local vs global properties GLOBAL: All software is the same. Lessons from one project directly apply to others. LOCAL: No software is the same: → (my Javascript startup, dev team, and business context are different than your Javascript startup, dev team, and business context). Lessons are not transferable. Tim Menzies and others showed you can in fact transfer (software) lessons.

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Software hierarchies • Look at data as having levels/hierarchies: → Radon samples by county, students per school → Ecosystems: Projects: Files: Methods • Incredibly useful when one “pool” is under-represented in dataset • This case: predicting mean log CBO for a given Spring project (to set thresholds) (files/projects)

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Bayesian regression Bayesian approach models a posterior distribution by conditioning on data, given some explicit prior assumptions. → Posterior P(H | D) is proportional to the likelihood P(D|H) times the prior P(D). → A machine for producing posterior distributions. More data = tighter estimates. Less data = more influence from priors.

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Hierarchical models Unpooled, linear regression y ⇠ N(µ, ) AAACCXicbVDLSsNAFJ3UV62vqEs3g61QQUrSjS6LblxJBfuAJpTJdNIOnZmEmYkQQrdu/BU3LhRx6x+482+ctFlo9cCFwzn3cu89Qcyo0o7zZZVWVtfWN8qbla3tnd09e/+gq6JEYtLBEYtkP0CKMCpIR1PNSD+WBPGAkV4wvcr93j2RikbiTqcx8TkaCxpSjLSRhjaspdBTlEOPIz3BiGU3s7rHkzMjjjk6rQ3tqtNw5oB/iVuQKijQHtqf3ijCCSdCY4aUGrhOrP0MSU0xI7OKlygSIzxFYzIwVCBOlJ/NP5nBE6OMYBhJU0LDufpzIkNcqZQHpjO/Vy17ufifN0h0eOFnVMSJJgIvFoUJgzqCeSxwRCXBmqWGICypuRXiCZIIaxNexYTgLr/8l3SbDddpuLfNauuyiKMMjsAxqAMXnIMWuAZt0AEYPIAn8AJerUfr2Xqz3hetJauYOQS/YH18A/HTmTM= 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Slide 8

Slide 8 text

Hierarchical models Unpooled, linear regression y ⇠ N(µ, ) AAACCXicbVDLSsNAFJ3UV62vqEs3g61QQUrSjS6LblxJBfuAJpTJdNIOnZmEmYkQQrdu/BU3LhRx6x+482+ctFlo9cCFwzn3cu89Qcyo0o7zZZVWVtfWN8qbla3tnd09e/+gq6JEYtLBEYtkP0CKMCpIR1PNSD+WBPGAkV4wvcr93j2RikbiTqcx8TkaCxpSjLSRhjaspdBTlEOPIz3BiGU3s7rHkzMjjjk6rQ3tqtNw5oB/iVuQKijQHtqf3ijCCSdCY4aUGrhOrP0MSU0xI7OKlygSIzxFYzIwVCBOlJ/NP5nBE6OMYBhJU0LDufpzIkNcqZQHpjO/Vy17ufifN0h0eOFnVMSJJgIvFoUJgzqCeSxwRCXBmqWGICypuRXiCZIIaxNexYTgLr/8l3SbDddpuLfNauuyiKMMjsAxqAMXnIMWuAZt0AEYPIAn8AJerUfr2Xqz3hetJauYOQS/YH18A/HTmTM= 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 Partial pooled regression Mean (log) CBO of Controller in project i alpha = intercept, Beta slope Slope and intercept vary per project y ⇠ N(↵j[i] + j[i] xi, ) AAACznichVJda9swFJW9ry77SrvHvYglgXgbwc7L+lIoXR8Cg+HB0hZiY64VOVEr2Z4ll3qq2et+3972A/Y/JqVmH21hFwRH55wr3aurtORMKt//4bh37t67/2DrYe/R4ydPn/W3d45kUVeEzknBi+okBUk5y+lcMcXpSVlRECmnx+nZO6sfn9NKsiL/pJqSxgJWOcsYAWWopP9zNGxwJJnAkQC1JsD1h3YcifqNIVcCvGFvNAzHM3yJDz28h6OsAqJxOD40zMzDrwycea02hNdar0m1tpQqSAL8ukNTfJEwKzcJszLwcg2JPl2wuP1t6rYXiWaGjGgpGTc1mrxba/zvGX9aSPoDf+JvAt8EQQcGqIsw6X+PlgWpBc0V4SDlIvBLFWuoFCOctr2olrQEcgYrujAwB0FlrDfjaPHIMEucFZVZucIb9u8MDULKRqTGafuR1zVL3qYtapXtxprlZa1oTq4uymqOVYHtbPGSVZQo3hgApGKmVkzWYAamzA+wjxBcb/kmOJpOAn8SfJwO9g+659hCL9BLNEYBeov20QyFaI6I89757HxxtBu6527rfr2yuk6X8xz9E+63X9e81qc= 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Src: https://bit.ly/2G04hN2

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Src: https://bit.ly/2G04hN2 Pooled

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Src: https://bit.ly/2G04hN2 Pooled Un-Pooled

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Src: https://bit.ly/2G04hN2 Pooled Un-Pooled Partially pooled

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Conditioning Finding closed form solutions is difficult or impossible Simulate using Hamiltonian Monte Carlo with a probabilistic program (Stan) Model checking vital: does our model converge

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Partial pooling model model { mu_a ~ normal(0, 100); # hyperprior mu_b ~ normal(0, 100); # hyperprior a ~ normal(mu_a, sigma_a); # prior b ~ normal(mu_b, sigma_b); # prior # likelihood: y ~ normal(a[project] + b[project]*x, sigma); }

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Validation: RMSE per project up to 50% lower with partial pooling

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Modeling and stating explicit priors (theories) increase replicability

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Modeling and stating explicit priors (theories) increase replicability Tooling support for Bayesian hierarchical models (Stan, PyMC3, Edward)

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Modeling and stating explicit priors (theories) increase replicability Tooling support for Bayesian hierarchical models (Stan, PyMC3, Edward) Partial pooling matches intuition of software development analysis

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Modeling and stating explicit priors (theories) increase replicability Tooling support for Bayesian hierarchical models (Stan, PyMC3, Edward) Partial pooling matches intuition of software development analysis “multi-level modeling deserves to be the default form of regression”

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Modeling and stating explicit priors (theories) increase replicability Tooling support for Bayesian hierarchical models (Stan, PyMC3, Edward) Partial pooling matches intuition of software development analysis More info: neilernst.net and @neilernst ArXiv: https://arxiv.org/abs/1804.02443 Jupyter notebook: https://doi.org/10.6084/m9.figshare.4892852.v1 Dataset: http://www.mauricioaniche.com/scam2016/ “multi-level modeling deserves to be the default form of regression”