Skip to content
Glacius
StatisticsConcept reference

Continuous likelihoods

A continuous likelihood evaluates the joint density at fixed observed points as a function of the candidate parameter.

2 min read1 figureSources & further reading
On this page 9 sections
  1. Overview
  2. Understand the idea
  3. An exact point has probability 0 under a continuous density
  4. Support matters
  5. Use the same observed values and measurement units when comparing candidates
  6. Key takeaway
  7. Further questions
  8. Sources & further reading
  9. Concept connections

01Understand the idea#

For a continuous model, an exact observed point has probability zero, yet its density can compare candidate parameter settings. A continuous likelihood multiplies the candidate densities at the observed values. It is a density-based score for the fixed sample, not the probability of those exact points.

For the same two independent observations, candidate A supplies densities 2 and 3; B supplies 1 and 4. Their likelihoods are 6 and 4. A has the larger density likelihood.

For one fixed independent two-observation sample, proper candidate density models supply values A:(2,3) and B:(1,4). The density products are 6 and 4 respectively. This is a table of evaluated densities and their products; entries are not point probabilities or shaded areas.For one fixed independent two-observation sample, proper candidate density models supply values A:(2,3) and B:(1,4). The density products are 6 and 4 respectively. This is a table of evaluated densities and their products; entries are not point probabilities or shaded areas.
Figure 1For one fixed independent two-observation sample, proper candidate density models supply values A:(2,3) and B:(1,4). The density products are 6 and 4 respectively. This is a table of evaluated densities and their products; entries are not point probabilities or shaded areas.
Link to this figure ↗Download SVGDownload PNG

Keep the observed points fixed and evaluate each candidate density at them. Independence lets those density values multiply.

L(θ)=∏ifθ(xi)L(\theta)=\prod_i f_\theta(x_i)

If a candidate gives densities 0.5 and 2 at two independent observations, their likelihood product is 1. Another candidate giving 0.8 and 0.8 has product 0.64. A density or product above 1 is allowed; total probability is controlled by integrating density over a region.

Check your reasoning

Fixed independent sample. Densities in data order: A (0.5,3)(0.5,3); B (1,1)(1,1). Give (LA,LB)(L_A,L_B).

  1. A(3.5,2)
  2. B(1.5,1)
  3. C(0.6,0.4)
Show answer and explanation
(1.5,1)

Products: (1.5,1)(1.5,1).

02An exact point has probability 0 under a continuous density#

An exact point has probability 0 under a continuous density. That does not set its density value to 0. Density likelihoods can exceed 1 and need not sum to 1 across candidates.

Check your reasoning

Independent data. Densities: A (0.2,2)(0.2,2); B (0.4,2)(0.4,2). Claim: zero point probabilities make likelihoods zero. Repair.

  1. A(0.4,0.8)
  2. B(0,0)
  3. C(2.2,2.4)
Show answer and explanation
(0.4,0.8)

Products: (0.4,0.8)(0.4,0.8).

03Support matters#

Support matters. With iid data (1,3)(1,3) and uniform density 1/θ1/\theta on 0<x<θ0<x<\theta, candidate θ=2\theta=2 gets likelihood 0 because 3 is outside. At θ=4\theta=4, both points are inside and the product is 1/161/16.

Check your reasoning

Ordered iid (0.1,0.3)(0.1,0.3). Density fθ(x)=1/θf_\theta(x)=1/\theta for 0<x<θ0<x<\theta, zero outside. θ=(0.5,2)\theta=(0.5,2). Give likelihoods.

  1. A(2,0.5)
  2. B(0,0)
  3. C(4,0.25)
Show answer and explanation
(4,0.25)

Products: (4,0.25)(4,0.25).

04Use the same observed values and measurement units when comparing candidates#

Use the same observed values and measurement units when comparing candidates. Changing units rescales densities and their products; it does not turn likelihood into a posterior distribution over parameter settings. Only relative comparisons within the same setup are intended here.

Key takeaway

Multiply all supplied densities after checking support. Density likelihood is a score for fixed observations, not their exact-point probability.

  • Compare candidate parameters using a fixed continuous sample density.

Further questions

Why take logs of a likelihood?
For positive factors, the logarithm turns their product into a sum while preserving likelihood order. It does not normalize the candidates.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Continuous likelihoods.” Math behind ML. /learn/s-continuous-likelihood