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This ratio tells us everything we need to know about the support the data lends to one hypothesis vis-a-vis the other. To quantify the statistical evidence for the first hypothesis against the second, we simply divide one probability by the other. 75, the probability of 6 heads in 10 tosses is: If the coin were a trick coin, so that p(heads) =. Where x is the number of heads obtained, n is the total number of flips, p is the probability of heads, and

5, the probability of this occurrence is defined by the binomial distribution: Suppose I flip a coin 10 times and it comes up 6 heads and 4 tails. The easiest way to explain this aspect of likelihood is to use the binomial distribution as an example. Only by comparing likelihoods do they become interpretable, because the constant in each likelihood cancels the other one out. Unlike a probability, a likelihood has no real meaning per se due to the arbitrary constant. They can be meaningful for planning studies or for decision analysis, but they are separate from the strength of the statistical evidence. Other facets of the data that do not factor into the likelihood function are irrelevant to the evaluation of the strength of the statistical evidence (Edwards, 1992, p. The Likelihood Principle states that the likelihood function contains all of the information relevant to the evaluation of statistical evidence. Furthermore, the strength of the statistical evidence for H1 over H2 is quantified by the ratio of their likelihoods, L(H1|D)/L(H2|D) (which again is proportional to P(D|H1)/P(D|H2) up to an arbitrary constant that cancels out). If these two probabilities are equivalent, then there is no evidence for either hypothesis over the other. That is, D is evidence for H1 over H2 if P(D|H1) > P(D|H2). In other words, there is evidence for H1 vis-a-vis H2 if and only if the probability of the data under H1 is greater than the probability of the data under H2. The Law of Likelihood states that “within the framework of a statistical model, a particular set of data supports one statistical hypothesis better than another if the likelihood of the first hypothesis, on the data, exceeds the likelihood of the second hypothesis” (Emphasis original. 30) defines the Likelihood Axiom as a natural combination of the Law of Likelihood and the Likelihood Principle.

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For conditional probability, the hypothesis is treated as a given and the data are free to vary. For likelihood, the data are a given and the hypotheses vary. The distinction is subtle, so I’ll say it again.

The likelihood of a hypothesis, L(H|D), conditions on the data as if they are fixed while allowing the hypotheses to vary. In the case of a conditional probability, P(D|H), the hypothesis is fixed and the data are free to vary. For example, likelihood need not sum to 1.Ī critical difference between probability and likelihood is in the interpretation of what is fixed and what can vary. Since a likelihood isn’t actually a probability it doesn’t obey various rules of probability. The likelihood of a hypothesis (H) given some data (D) is proportional to the probability of obtaining D given that H is true, multiplied by an arbitrary positive constant (K). It’s not a probability, but it is proportional to a probability. In order to understand Bayesian model comparison (Bayes factors) you need to understand the likelihood and likelihood ratios. In order to understand Bayesian parameter estimation you need to understand the likelihood. The likelihood is the workhorse of Bayesian inference. One thing that often gets left out of the discussion is the importance of the likelihood. Should we embrace priors, or should we be skeptical? When are Bayesian methods sensitive to specification of the prior, and when do the data effectively overwhelm it? Should we use context specific prior distributions or should we use general defaults? These are all great questions and great discussions to be having. Much of the discussion in psychology surrounding Bayesian inference focuses on priors.

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