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I think you misunderstand the concept of conditional probability, when we sample from the prior and plug the sampled value into the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .In the following is an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

I think you misunderstand the concept of conditional probability, when we sample from the prior and plug the sampled value into the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .In the following is an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

I think you misunderstand the concept of conditional probability, when we sample from the prior and plug the sampled value into the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

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I think you misunderstand the concept of conditional probability, when we sample from the prior and plug in the sampled value ininto the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .Anyway I want to give youIn the following is an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

For more details see A First Course in Bayesian Statistical Methods the book provides also a very good comparison between MC and MCMC Chapter 6 section "Introduction to MCMC diagnostics ".

I think you misunderstand the concept of conditional probability, when we sample from the prior and plug in the sampled value in the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .Anyway I want to give you an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

I think you misunderstand the concept of conditional probability, when we sample from the prior and plug the sampled value into the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .In the following is an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

For more details see A First Course in Bayesian Statistical Methods the book provides also a very good comparison between MC and MCMC Chapter 6 section "Introduction to MCMC diagnostics ".

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I think you misunderstand the concept of conditional probability, when we sample formfrom the prior and plug in the sampled value in the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .Anyway I want to give you an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

I think you misunderstand the concept of conditional probability, when we sample form the prior and plug in the sampled value in the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .Anyway I want to give you an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

I think you misunderstand the concept of conditional probability, when we sample from the prior and plug in the sampled value in the likelihood , we do not evaluate a probability value at a particular point we just get a conditional distribution given the sampled value from the prior .Anyway I want to give you an example from a frequentist point of view but it is very straightforward to be understandable in the bayesian context. Suppose that the target distribution is the joint probability distribution of two variables, a discrete variable $\delta \in \{1,2,3\}$ and a continuous variable $\theta \in \mathbb{R}$ then the target density will be defined as :

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