Likelihood Calculator

Calculate the likelihood of observed data given a set of model parameters.

About Likelihood Calculator

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

Worked example

With the default values below, Likelihood Calculator returns Log-Likelihood of -2.7090. Change any input above and every figure here is recalculated in your browser.

Inputs used

Observed Values: 4.2, 5.1, 4.8, 5.3, 4.9
Assumed Mean (μ): 5
Assumed Std Dev (σ)minimum 0.001: 0.5

Results

Log-Likelihood: -2.7090
Likelihood: 0.06660626
Observations: 5

Step by step

1Number of observations= 5
2Log-likelihood (Gaussian): Σ[-0.5ln(2πσ²) − (x−μ)²/(2σ²)]= -2.708957
3Likelihood: exp(log-likelihood)= 6.6606e-2

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

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.