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The fiducial argument is to interpret likelihood as a probability. Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.

Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant with likelihood function $\lambda^n \prod_{i=1}^n e^{-\lambda x_i} = \lambda^n e^{-\lambda(x_1+\ldots+x_n)}$.

In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. For instance, if my prior belief about the value of lambda is well represented by the density distribution $2.3 \cdot e^{-2.3\lambda}$, the joint distribution is the product of the two, i.e. $2.3 \cdot \lambda^n e^{-\lambda(2.3+x_1+\ldots+x_n) }$. The posterior is the distribution of $\lambda$ given the measurements, which is computed with Bayes formula. In this case, $\lambda$ has a Gamma distribution with parameters $n$ and $2.3+x_1+\ldots+x_n$.

In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$. To follow up on the example above, the fiducial distribution is $\lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. This is the same as the likelihood, except that it is now interpreted as a probability. With proper scaling, it is a Gamma distribution with parameters $n$ and $x_1+\ldots+x_n$.

Those differences have most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).

The fiducial argument is to interpret likelihood as a probability.
Edit: not exactly, see the answer of @Sextus for more details.

Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.

Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant with likelihood function $\lambda^n \prod_{i=1}^n e^{-\lambda x_i} = \lambda^n e^{-\lambda(x_1+\ldots+x_n)}$.

In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. For instance, if my prior belief about the value of lambda is well represented by the density distribution $2.3 \cdot e^{-2.3\lambda}$, the joint distribution is the product of the two, i.e. $2.3 \cdot \lambda^n e^{-\lambda(2.3+x_1+\ldots+x_n) }$. The posterior is the distribution of $\lambda$ given the measurements, which is computed with Bayes formula. In this case, $\lambda$ has a Gamma distribution with parameters $n$ and $2.3+x_1+\ldots+x_n$.

In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$. To follow up on the example above, the fiducial distribution is $\lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. This is the same as the likelihood, except that it is now interpreted as a probability. With proper scaling, it is a Gamma distribution with parameters $n$ and $x_1+\ldots+x_n$.

Those differences have most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).

The fiducial argument is to interpret likelihood as a probability. Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.

Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant with likelihood function $\lambda^n \prod_{i=1}^n e^{-\lambda x_i} = \lambda^n e^{-\lambda(x_1+\ldots+x_n)}$.

In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. For instance, if my prior belief about the value of lambda is well represented by the density distribution $2.3 \cdot e^{-2.3\lambda}$, the joint distribution is the product of the two, i.e. $2.3 \cdot \lambda^n e^{-\lambda(2.3+x_1+\ldots+x_n) }$. The posterior is the distribution of $\lambda$ given the measurements, which is computed with Bayes formula. In this case, $\lambda$ has a Gamma distribution with parameters $n$ and $2.3+x_1+\ldots+x_n$.

In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$. To follow up the example above, the fiducial distribution is $\lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. This is the same as the likelihood, except that it is now interpreted as a probability. With proper scaling, it is a Gamma distribution with parameters $n$ and $x_1+\ldots+x_n$.

Those differences have most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).

The fiducial argument is to interpret likelihood as a probability. Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.

Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant with likelihood function $\lambda^n \prod_{i=1}^n e^{-\lambda x_i} = \lambda^n e^{-\lambda(x_1+\ldots+x_n)}$.

In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. For instance, if my prior belief about the value of lambda is well represented by the density distribution $2.3 \cdot e^{-2.3\lambda}$, the joint distribution is the product of the two, i.e. $2.3 \cdot \lambda^n e^{-\lambda(2.3+x_1+\ldots+x_n) }$. The posterior is the distribution of $\lambda$ given the measurements, which is computed with Bayes formula. In this case, $\lambda$ has a Gamma distribution with parameters $n$ and $2.3+x_1+\ldots+x_n$.

In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$. To follow up on the example above, the fiducial distribution is $\lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. This is the same as the likelihood, except that it is now interpreted as a probability. With proper scaling, it is a Gamma distribution with parameters $n$ and $x_1+\ldots+x_n$.

Those differences have most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).

The fiducial argument is to interpret likelihood as a probability. Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.

Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant. In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$.

This has most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).

The fiducial argument is to interpret likelihood as a probability. Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.

Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant with likelihood function $\lambda^n \prod_{i=1}^n e^{-\lambda x_i} = \lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. 

In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. For instance, if my prior belief about the value of lambda is well represented by the density distribution $2.3 \cdot e^{-2.3\lambda}$, the joint distribution is the product of the two, i.e. $2.3 \cdot \lambda^n e^{-\lambda(2.3+x_1+\ldots+x_n) }$. The posterior is the distribution of $\lambda$ given the measurements, which is computed with Bayes formula. In this case, $\lambda$ has a Gamma distribution with parameters $n$ and $2.3+x_1+\ldots+x_n$. 

In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$. To follow up the example above, the fiducial distribution is $\lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. This is the same as the likelihood, except that it is now interpreted as a probability. With proper scaling, it is a Gamma distribution with parameters $n$ and $x_1+\ldots+x_n$.

Those differences have most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).