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Cocalc Section5 2 Bayesian Optimization Ipynb

Leo Migdal
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cocalc section5 2 bayesian optimization ipynb

A critical task in most machine learning or probabilistic programming pipelines is the optimization of model hyperparameters. Several strategies can be used for function optimization, such as randomly sampling the parameter space (random search) or systematically evaluating the parameter space (grid search). This is often not trivial, because the loss function for a particular parameter can be noisy and non-linear, and for most problems we are omptimizing a set of parameters simultaneously, which can result in... Moreover, for large problems and complex models (e.g. deep neural networks) a single model run can be expensive and time-consuming. As a result, doing systematic searches over the hyperparameter space is infeasible, and random searches are usually ineffective.

To circumvent this, Bayesian optimization offers a principled and efficient approach for directing a search of arbitrary global optimization problems. It involves constructing a probabilistic model of the objective function, and then using an auxiliary function, called an acquisition function, to obtain candidate values for evaluation using the true objective function. Bayesian Optimization is often used in applied machine learning to tune the hyperparameters of a given model on a validation dataset. Global function optimization involves finding the minimum (maximum) of a function of interest. Samples are drawn from the domain and evaluated by the objective function to give a score or cost. These samples are candidate optimal values, which are compared to previous samples based on their cost.

While the objective function may be simple to specify mathematically and in code, it can be computationally challenging to compute, and its form may be non-linear and multi-dimensional. Moreover, its solution may be non-convex, implying that a discovered mimimum value may not be a global minimum. Specific to data science, many machine learning algorithms involve the optimization of weights, coefficients, and hyperparameters based on information contained in training data. We seek a principled method for evaluating the parmaeter space, such that consecutive samples are taken from regions of the search space that are more likely to contain minima. There was an error while loading. Please reload this page.

Lecture slides for UCLA LS 30B, Spring 2020 Be able to explain the significance of optimization in biology, and give several examples. Be able to describe the main biological process that underlies all optimization problems in biology. Be able to distinguish local maxima and local minima of a function from global extrema. Know the significance of local maxima in evolution. In order to understand how Bayesian statistics works, we're going to start with a simple scenario.

Imagine you are studying coastal birds of southern California. Two models describing the abundance of seagulls and cormorants have been proposed. Model 1 predicts that you will observe 75% cormorants and 25% seagulls, while model 2 predicts that each species will make up 50% of your observations. Heading out to Ballona Creek to observe birds, you want to know the probability that model 1 is true. Let's call "model 1 being true" event A and "observing a bird (either seagull or cormorant)" event B. We can easily find p(B∣A)p(B|A)p(B∣A), but using this information to determine the probability that model 1 is correct requires an equation known as Bayes' theorem.

In this lab, you will first compute p(A∣B)p(A|B)p(A∣B) and then update this probability based on new evidence (observing more birds). Set up an array for each model with the probabilities of observing cormorants and seagulls. If you know nothing that would favor one model over the other, what is the probability that model 1 is best? Set up another array that has the probabilities of each model. 📚 The CoCalc Library - books, templates and other resources In this section we will discuss how Bayesians think about data, and how we can estimate model parameters using a technique called MCMC.

When I started to learn how to apply Bayesian methods, I found it very useful to understand how Bayesians think about data. Imagine the following scenario: A curious boy watches the number of cars that pass by his house every day. He diligently notes down the total count of cars that pass per day. Over the past week, his notebook contains the following counts: 12, 33, 20, 29, 20, 30, 18 From a Bayesian's perspective, this data is generated by a random process.

However, now that the data is observed, it is fixed and does not change. This random process has some model parameters that are fixed. However, the Bayesian uses probability distributions to represent his/her uncertainty in these parameters. 📚 The CoCalc Library - books, templates and other resources Previously, we addressed the question: "is my chat response time effected by who I'm talking to?". We have estimated model parameters for each individual I've had conversations with.

But sometimes we want to understand the effect of more factors such as "day of week," "time of day," etc. We can use GLM (generalized linear models) to better understand the effects of these factors. When we have a response yyy that is continuous from −∞-\infty−∞ to ∞\infty∞, we can consider using a linear regression represented by: We read this as: our response is normally distributed around μ\muμ with a standard deviation of σ\sigmaσ. The value of μ\muμ is described by a linear function of explanatory variables XβX \betaXβ with a baseline intercept β0\beta_0β0​. In the event you're not modeling a continuous response variable from −∞-\infty−∞ to ∞\infty∞, you may need to use a link function to transform your response range.

For a Poisson distribution, the canonical link function used is the log link. This can be formally described as: 📚 The CoCalc Library - books, templates and other resources

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