Likelihood Calculator
Calculate the likelihood of observed data given a set of model parameters.
Inputs
Comma-separated data points
Log-Likelihood
-2.7090
Likelihood
0.06660626
Observations
5
Step by step
Number of observations
n
= 5
Log-likelihood (Gaussian): Σ[-0.5ln(2πσ²) − (x−μ)²/(2σ²)]
n=5, μ=5, σ=0.5
= -2.708957
Likelihood: exp(log-likelihood)
e^(-2.7090)
= 6.6606e-2
How it works
The likelihood function measures how probable the observed data is under a given set of model parameters. For Gaussian-distributed data, the log-likelihood is the sum of log-probabilities of each observation under the assumed normal distribution N(μ, σ²). Maximum Likelihood Estimation (MLE) finds parameters that maximize this value.
Formula
Gaussian Log-Likelihood
LL = sum(-0.5*ln(2*pi*sigma^2) - (x_i - mu)^2 / (2*sigma^2))
- mu
- Assumed population mean
- sigma
- Assumed standard deviation
Frequently Asked Questions
Why use log-likelihood instead of likelihood?
Multiplying many small probabilities leads to numerical underflow. Taking logs converts products to sums, which is numerically stable and mathematically equivalent for optimization (maximizing log-likelihood maximizes likelihood).
How does this relate to training neural networks?
Training with cross-entropy loss is equivalent to maximizing the log-likelihood of the training data under the model's predicted distribution — MLE is the statistical foundation of most supervised learning.