Log probability

As explained in Math and Architectures of Deep Learning

The natural logarithm of a probability density or mass value, used to make products and tiny probabilities easier to compute with.

Log probability is the logarithm of a probability or density value. Machine learning commonly uses it because exponential probability expressions become simpler after taking logarithms. section 5.9 section 6.6.1

The log transform is useful because it preserves the maximizing value while changing products into sums:

• log(e^x) = x section 6.6.1
• log(Π e^x(i)) = Σ x(i), turning a product of exponential terms into an additive expression. section 6.6.1
• Maximizing likelihood and maximizing log-likelihood produce the same parameter values, while the logarithmic form simplifies the optimization. section 6.6.1 section 6.6.2

Gaussian maximum likelihood

For adult female height samples, maximizing the Gaussian log-likelihood with respect to the mean and variance yields the sample mean and variance. Those estimates define a Gaussian that can assign a probability to a new height. section 6.8

Equation 6.20

Inspect the likelihood expression and its logarithmic optimization counterpart. section 6.6.1

How Math and Architectures of Deep Learning covers it

How deep it goes

Explains the product-to-sum identity and optimization rationale, then applies them to Gaussian estimation without deriving every distribution internally.

Examples

Uses exponential identities, Gaussian likelihood optimization, adult height samples, and PyTorch distribution calculations.

What it assumes

Assumes familiarity with probability distributions, exponential functions, and likelihood-based parameter estimation.

What you can do after

Apply it via PyTorch distributions and optimize Gaussian or other exponential-family likelihoods in logarithmic form.

How it connects

Connects probability distributions, likelihood, maximum likelihood estimation, Gaussian distributions, and Gaussian mixture models.

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Also explained in

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  Build a Reasoning Model (From Scratch)The logarithm of a model-assigned probability, used here to score how likely a generated token or whole sequence is under the current parameters.
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  Statistics Every Programmer NeedsLog-likelihood is the logarithm of the likelihood of observed data under a statistical model and its parameter values.
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  Machine Learning Algorithms in DepthA log likelihood is the logarithm of a model likelihood for observed data, commonly optimized instead of the likelihood itself.
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  Build AI-Enhanced Web AppsA log probability is the logarithm of the probability assigned to an event, such as a language-model token.
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  Generative AI in ActionLogprobs are completion metadata containing the natural logarithms of the model probabilities assigned to candidate tokens.
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  Causal AIA log probability is the natural logarithm of a probability value.
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  Reinforcement Learning from Human FeedbackA log probability is the logarithm of the probability assigned to an event or token by a probabilistic model.

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Related concepts

• policy gradient methodused by · A policy gradient method is a reinforcement-learning technique that uses gradients of expected return to optimize the parameters of a policy.
• generationpart of · Generation is the computational production of outputs from supplied inputs or context, often by applying a model to produce predictions or new content.
• completions apipart of · A completions API is an interface for requesting language-model-generated text from supplied prompts or inputs and configuring generation behavior.
• McFadden's pseudo R-squaredused by · McFadden's pseudo R-squared is a likelihood-based goodness-of-fit measure comparing a fitted discrete-choice model with an intercept-only null model.
• policyuses · A policy is a rule, strategy, or mapping that governs or determines decisions, actions, permissions, or behavior under specified conditions.

This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.