Tags | Peter’s Notes

Tags starting with A:

A/B-testing:

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.
The Beta Distribution: A Distribution Over Probabilities
14 May 2026
Understand the Beta distribution as a model for unknown success probabilities, and derive the Beta-Bernoulli update for binary data.
Multi-armed bandit implementation
26 Sep 2018
How to implement a Bayesian multi-armed bandit model in Python, and use it to simulate an online test. The model is based on the beta distribution and Thompson sampling.

Agents:

Implement a simple ReAct Agent using OpenAI function calling
21 Jan 2024
Simple example of using OpenAI Function Calling in a ReAct Loop to execute simple multistep tasks.
ReAct REPL Agent
07 Apr 2023
GPT-Based ReAct Python REPL Agent with access to method retrieval. Can execute simple workflows by chaining API calls.

Tags starting with B:

Backpropagation:

How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.
How to implement a neural network (5/5) - Generalization to multiple layers
16 Jun 2015
Generalization of neural networks to multiple layers. Illustrated on a simple network build from scratch using Python and NumPy. The network is trained on a digit classification toy problem using stochastic gradient descent.
How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.
How to implement a neural network (3/5) - backpropagation
14 Jun 2015
Transition from single-layer linear models to a multi-layer neural network by adding a hidden layer with a nonlinearity. A minimal network is implemented using Python and NumPy. This minimal network is simple enough to visualize its parameter space. The model will be optimized on a toy problem using backpropagation and gradient descent, for which the gradient derivations are included.

Bayesian Inference:

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.
The Beta Distribution: A Distribution Over Probabilities
14 May 2026
Understand the Beta distribution as a model for unknown success probabilities, and derive the Beta-Bernoulli update for binary data.

Bernoulli Distribution:

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.
The Beta Distribution: A Distribution Over Probabilities
14 May 2026
Understand the Beta distribution as a model for unknown success probabilities, and derive the Beta-Bernoulli update for binary data.

Beta Distribution:

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.
The Beta Distribution: A Distribution Over Probabilities
14 May 2026
Understand the Beta distribution as a model for unknown success probabilities, and derive the Beta-Bernoulli update for binary data.
Multi-armed bandit implementation
26 Sep 2018
How to implement a Bayesian multi-armed bandit model in Python, and use it to simulate an online test. The model is based on the beta distribution and Thompson sampling.

Beta-Binomial Distribution:

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.

Binomial Distribution:

The Beta Distribution: A Distribution Over Probabilities
14 May 2026
Understand the Beta distribution as a model for unknown success probabilities, and derive the Beta-Bernoulli update for binary data.

Blog:

Tagging GitHub Pages
15 May 2021
How to use Jekyll tags on GitHub Pages blogs. How to sort tags using liquid templating.
Hosting Jupyter Notebooks on a Blog
15 May 2021
How this blog, hosted on GitHub Pages, uses Jekyll and Jupyter Notebooks to view notebooks as blog posts.

Tags starting with C:

Classification:

How to implement a neural network (5/5) - Generalization to multiple layers
16 Jun 2015
Generalization of neural networks to multiple layers. Illustrated on a simple network build from scratch using Python and NumPy. The network is trained on a digit classification toy problem using stochastic gradient descent.
How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.
How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.
Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Cross-Entropy:

How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.
Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Tags starting with D:

Diffusion:

Flow Matching: A visual introduction
01 Nov 2025
Visual introduction to flow matching. Illustrates the flow matching model, the velocity field, and the sampled paths using a simple 1D toy example.

Tags starting with F:

Flow Matching:

Flow Matching: A visual introduction
01 Nov 2025
Visual introduction to flow matching. Illustrates the flow matching model, the velocity field, and the sampled paths using a simple 1D toy example.

Tags starting with G:

Gaussian Distribution:

Gaussian processes (1/3) - From scratch
05 Jan 2019
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.
Multivariate normal distribution
28 Sep 2018
Introduction to the multivariate normal distribution (Gaussian). We'll describe how to sample from this distribution and how to compute its conditionals and marginals.

Gaussian Process:

Gaussian processes (3/3) - exploring kernels
07 Jan 2019
Explore the Gaussian process kernels fitted by the previous post by using various visualizations.
Gaussian processes (2/3) - Fitting a Gaussian process kernel
06 Jan 2019
Fit a parameterized Gaussian process kernel on the Mauna Loa CO₂ dataset. We'll use JAX and Optax to implement the model and fit the kernel parameters.
Gaussian processes (1/3) - From scratch
05 Jan 2019
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.

Generative Models:

Flow Matching: A visual introduction
01 Nov 2025
Visual introduction to flow matching. Illustrates the flow matching model, the velocity field, and the sampled paths using a simple 1D toy example.

GitHub Pages:

Tagging GitHub Pages
15 May 2021
How to use Jekyll tags on GitHub Pages blogs. How to sort tags using liquid templating.
Hosting Jupyter Notebooks on a Blog
15 May 2021
How this blog, hosted on GitHub Pages, uses Jekyll and Jupyter Notebooks to view notebooks as blog posts.

Gradient Descent:

Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.
How to implement a neural network (5/5) - Generalization to multiple layers
16 Jun 2015
Generalization of neural networks to multiple layers. Illustrated on a simple network build from scratch using Python and NumPy. The network is trained on a digit classification toy problem using stochastic gradient descent.
How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
How to implement a neural network (1/5) - gradient descent
12 Jun 2015
How to implement, and optimize, a linear regression model from scratch using Python and NumPy. The linear regression model will be approached as a minimal regression neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.
Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Gradient checking:

How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.

Tags starting with J:

JAX:

Gaussian processes (3/3) - exploring kernels
07 Jan 2019
Explore the Gaussian process kernels fitted by the previous post by using various visualizations.
Gaussian processes (2/3) - Fitting a Gaussian process kernel
06 Jan 2019
Fit a parameterized Gaussian process kernel on the Mauna Loa CO₂ dataset. We'll use JAX and Optax to implement the model and fit the kernel parameters.

Jekyll:

Tagging GitHub Pages
15 May 2021
How to use Jekyll tags on GitHub Pages blogs. How to sort tags using liquid templating.
Hosting Jupyter Notebooks on a Blog
15 May 2021
How this blog, hosted on GitHub Pages, uses Jekyll and Jupyter Notebooks to view notebooks as blog posts.

Tags starting with K:

Kernel:

Gaussian processes (3/3) - exploring kernels
07 Jan 2019
Explore the Gaussian process kernels fitted by the previous post by using various visualizations.
Gaussian processes (2/3) - Fitting a Gaussian process kernel
06 Jan 2019
Fit a parameterized Gaussian process kernel on the Mauna Loa CO₂ dataset. We'll use JAX and Optax to implement the model and fit the kernel parameters.
Gaussian processes (1/3) - From scratch
05 Jan 2019
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.

Tags starting with L:

LLM:

Implement a simple ReAct Agent using OpenAI function calling
21 Jan 2024
Simple example of using OpenAI Function Calling in a ReAct Loop to execute simple multistep tasks.
ReAct REPL Agent
07 Apr 2023
GPT-Based ReAct Python REPL Agent with access to method retrieval. Can execute simple workflows by chaining API calls.

Least Squares:

Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.

Linear Regression:

Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.
How to implement a neural network (1/5) - gradient descent
12 Jun 2015
How to implement, and optimize, a linear regression model from scratch using Python and NumPy. The linear regression model will be approached as a minimal regression neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.

Liquid Templating:

Tagging GitHub Pages
15 May 2021
How to use Jekyll tags on GitHub Pages blogs. How to sort tags using liquid templating.

Logistic Function:

Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Logistic Regression:

How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.
Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Tags starting with M:

MCMC:

Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.

Machine Learning:

Gaussian processes (2/3) - Fitting a Gaussian process kernel
06 Jan 2019
Fit a parameterized Gaussian process kernel on the Mauna Loa CO₂ dataset. We'll use JAX and Optax to implement the model and fit the kernel parameters.
Gaussian processes (1/3) - From scratch
05 Jan 2019
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.
Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.
How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.
How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.
How to implement a neural network (5/5) - Generalization to multiple layers
16 Jun 2015
Generalization of neural networks to multiple layers. Illustrated on a simple network build from scratch using Python and NumPy. The network is trained on a digit classification toy problem using stochastic gradient descent.
How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.
How to implement a neural network (3/5) - backpropagation
14 Jun 2015
Transition from single-layer linear models to a multi-layer neural network by adding a hidden layer with a nonlinearity. A minimal network is implemented using Python and NumPy. This minimal network is simple enough to visualize its parameter space. The model will be optimized on a toy problem using backpropagation and gradient descent, for which the gradient derivations are included.
How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
How to implement a neural network (1/5) - gradient descent
12 Jun 2015
How to implement, and optimize, a linear regression model from scratch using Python and NumPy. The linear regression model will be approached as a minimal regression neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.
Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Metropolis-Hastings:

Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.

Momentum:

How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.

Multi-Armed Bandit (MAB):

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.
Multi-armed bandit implementation
26 Sep 2018
How to implement a Bayesian multi-armed bandit model in Python, and use it to simulate an online test. The model is based on the beta distribution and Thompson sampling.

Tags starting with N:

Nesterov momentum:

How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.

Neural Networks:

Flow Matching: A visual introduction
01 Nov 2025
Visual introduction to flow matching. Illustrates the flow matching model, the velocity field, and the sampled paths using a simple 1D toy example.
How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.
How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.
How to implement a neural network (5/5) - Generalization to multiple layers
16 Jun 2015
Generalization of neural networks to multiple layers. Illustrated on a simple network build from scratch using Python and NumPy. The network is trained on a digit classification toy problem using stochastic gradient descent.
How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.
How to implement a neural network (3/5) - backpropagation
14 Jun 2015
Transition from single-layer linear models to a multi-layer neural network by adding a hidden layer with a nonlinearity. A minimal network is implemented using Python and NumPy. This minimal network is simple enough to visualize its parameter space. The model will be optimized on a toy problem using backpropagation and gradient descent, for which the gradient derivations are included.
How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
How to implement a neural network (1/5) - gradient descent
12 Jun 2015
How to implement, and optimize, a linear regression model from scratch using Python and NumPy. The linear regression model will be approached as a minimal regression neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.
Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Nonlinearity:

How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.
How to implement a neural network (3/5) - backpropagation
14 Jun 2015
Transition from single-layer linear models to a multi-layer neural network by adding a hidden layer with a nonlinearity. A minimal network is implemented using Python and NumPy. This minimal network is simple enough to visualize its parameter space. The model will be optimized on a toy problem using backpropagation and gradient descent, for which the gradient derivations are included.

Notebook:

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.
The Beta Distribution: A Distribution Over Probabilities
14 May 2026
Understand the Beta distribution as a model for unknown success probabilities, and derive the Beta-Bernoulli update for binary data.
Flow Matching: A visual introduction
01 Nov 2025
Visual introduction to flow matching. Illustrates the flow matching model, the velocity field, and the sampled paths using a simple 1D toy example.
Hosting Jupyter Notebooks on a Blog
15 May 2021
How this blog, hosted on GitHub Pages, uses Jekyll and Jupyter Notebooks to view notebooks as blog posts.
Gaussian processes (3/3) - exploring kernels
07 Jan 2019
Explore the Gaussian process kernels fitted by the previous post by using various visualizations.
Gaussian processes (2/3) - Fitting a Gaussian process kernel
06 Jan 2019
Fit a parameterized Gaussian process kernel on the Mauna Loa CO₂ dataset. We'll use JAX and Optax to implement the model and fit the kernel parameters.
Gaussian processes (1/3) - From scratch
05 Jan 2019
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.
Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.
Multivariate normal distribution
28 Sep 2018
Introduction to the multivariate normal distribution (Gaussian). We'll describe how to sample from this distribution and how to compute its conditionals and marginals.
Multi-armed bandit implementation
26 Sep 2018
How to implement a Bayesian multi-armed bandit model in Python, and use it to simulate an online test. The model is based on the beta distribution and Thompson sampling.
How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.
How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.
How to implement a neural network (5/5) - Generalization to multiple layers
16 Jun 2015
Generalization of neural networks to multiple layers. Illustrated on a simple network build from scratch using Python and NumPy. The network is trained on a digit classification toy problem using stochastic gradient descent.
How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.
How to implement a neural network (3/5) - backpropagation
14 Jun 2015
Transition from single-layer linear models to a multi-layer neural network by adding a hidden layer with a nonlinearity. A minimal network is implemented using Python and NumPy. This minimal network is simple enough to visualize its parameter space. The model will be optimized on a toy problem using backpropagation and gradient descent, for which the gradient derivations are included.
How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
How to implement a neural network (1/5) - gradient descent
12 Jun 2015
How to implement, and optimize, a linear regression model from scratch using Python and NumPy. The linear regression model will be approached as a minimal regression neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.
Logistic classification with cross-entropy (1/2)
10 Jun 2015
Description of the logistic function used to model binary classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

NumPy:

Gaussian processes (1/3) - From scratch
05 Jan 2019
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.
How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.
How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.
How to implement a neural network (5/5) - Generalization to multiple layers
16 Jun 2015
Generalization of neural networks to multiple layers. Illustrated on a simple network build from scratch using Python and NumPy. The network is trained on a digit classification toy problem using stochastic gradient descent.
How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.
How to implement a neural network (3/5) - backpropagation
14 Jun 2015
Transition from single-layer linear models to a multi-layer neural network by adding a hidden layer with a nonlinearity. A minimal network is implemented using Python and NumPy. This minimal network is simple enough to visualize its parameter space. The model will be optimized on a toy problem using backpropagation and gradient descent, for which the gradient derivations are included.
How to implement a neural network (2/5) - classification
13 Jun 2015
How to implement, and optimize, a logistic regression model from scratch using Python and NumPy. The logistic regression model will be approached as a minimal classification neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.
How to implement a neural network (1/5) - gradient descent
12 Jun 2015
How to implement, and optimize, a linear regression model from scratch using Python and NumPy. The linear regression model will be approached as a minimal regression neural network. The model will be optimized using gradient descent, for which the gradient derivations are provided.

Tags starting with O:

OpenAI:

Implement a simple ReAct Agent using OpenAI function calling
21 Jan 2024
Simple example of using OpenAI Function Calling in a ReAct Loop to execute simple multistep tasks.

Optax:

Gaussian processes (2/3) - Fitting a Gaussian process kernel
06 Jan 2019
Fit a parameterized Gaussian process kernel on the Mauna Loa CO₂ dataset. We'll use JAX and Optax to implement the model and fit the kernel parameters.

Tags starting with P:

Probability:

The Beta Distribution: A Distribution Over Probabilities
14 May 2026
Understand the Beta distribution as a model for unknown success probabilities, and derive the Beta-Bernoulli update for binary data.
Gaussian processes (1/3) - From scratch
05 Jan 2019
This post explores some concepts behind Gaussian processes, such as stochastic processes and the kernel function. We will build up deeper understanding of Gaussian process regression by implementing them from scratch using Python and NumPy.
Regression quattro stagioni
22 Oct 2018
Linear regression implemented in four different ways. We'll describe the model and four different ways of estimating its parameters: MLE, OLS, gradient descent, & MCMC.
Multivariate normal distribution
28 Sep 2018
Introduction to the multivariate normal distribution (Gaussian). We'll describe how to sample from this distribution and how to compute its conditionals and marginals.

Tags starting with R:

RMSProp:

How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.

RProp:

How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.

Radial Basis Function (RBF):

How to implement a neural network (3/5) - backpropagation
14 Jun 2015
Transition from single-layer linear models to a multi-layer neural network by adding a hidden layer with a nonlinearity. A minimal network is implemented using Python and NumPy. This minimal network is simple enough to visualize its parameter space. The model will be optimized on a toy problem using backpropagation and gradient descent, for which the gradient derivations are included.

ReAct Loop:

Implement a simple ReAct Agent using OpenAI function calling
21 Jan 2024
Simple example of using OpenAI Function Calling in a ReAct Loop to execute simple multistep tasks.
ReAct REPL Agent
07 Apr 2023
GPT-Based ReAct Python REPL Agent with access to method retrieval. Can execute simple workflows by chaining API calls.

Rectified Flow:

Flow Matching: A visual introduction
01 Nov 2025
Visual introduction to flow matching. Illustrates the flow matching model, the velocity field, and the sampled paths using a simple 1D toy example.

Recurrent Neural Network (RNN):

How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.
How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.

Tags starting with S:

Sampling:

Multivariate normal distribution
28 Sep 2018
Introduction to the multivariate normal distribution (Gaussian). We'll describe how to sample from this distribution and how to compute its conditionals and marginals.

Sequential Data:

Beta Priors for Self-Reinforcing Binary Decisions
17 May 2026
See how Beta-Bernoulli updates work one observation at a time in A/B tests and bandits, and how the Pólya urn explains self-reinforcing feedback.
How to implement an RNN (2/2) - Tensor data and non-linearities
27 Sep 2015
How to implement and train a simple recurrent neural network (RNN) with input data stored as a tensor. The RNN will be learning how to perform binary addition as a toy problem. RMSProp and Nesterov momentum are used as a gradient-based optimization algorithm during training.
How to implement an RNN (1/2) - Minimal example
27 Sep 2015
How to implement a minimal recurrent neural network (RNN) from scratch with Python and NumPy. The RNN is simple enough to visualize the loss surface and explore why vanishing and exploding gradients can occur during optimization. For stability, the RNN will be trained with backpropagation through time using the RProp optimization algorithm.

Softmax:

Softmax classification with cross-entropy (2/2)
11 Jun 2015
Description of the softmax function used to model multiclass classification problems. Contains derivations of the gradients used for optimizing any parameters with regards to the cross-entropy loss function.

Tags starting with V:

Vectorization:

How to implement a neural network (4/5) - vectorization of operations
15 Jun 2015
Vectorization of the neural network and backpropagation algorithm for multi-dimensional data. Vectorization of operations is illustrated on a simple network implemented using Python and NumPy. The network is trained on a toy problem using gradient descent with momentum.

Velocity Field:

Flow Matching: A visual introduction
01 Nov 2025
Visual introduction to flow matching. Illustrates the flow matching model, the velocity field, and the sampled paths using a simple 1D toy example.