# What does the likelihood of a model mean?

So, I understand the idea of likelihood when it comes to throwing dice. For example:

Denote two heads in two tosses HH. Assuming that each successive coin flip is i.i.d., then the probability of observing HH is

$$P({HH}\mid p_{\text{H}}=0.5)=0.5^2=0.25.$$

But when it comes to models I get really confused. To calculate the Bayes factor, we can use Bayes rule to find the likelihood of model $$M,$$ given the data $$D,$$ in this way: $$P(M\mid D)= \frac{P(D\mid M)×P(M)}{P(D)}$$

But what does this mean intuitively? Likelihood of a model doesn't make sense to me.

Another question: If I have two models $$M_1$$ and $$M_2$$ and $$M_1 \subseteq M_2$$ meaning that $$M_2$$ extend $$M_1$$, then we must always have that $$P(M_2\mid D) \geq P(M_1\mid D)$$, right? But then the probability odds $$\frac{P(M_1\mid D)}{P(M_2\mid D)} < 1$$

What do I miss? I hope you can clarify me a bit. All answers are appreciated. Thanks in advance.

Suppose the coin turns up "heads" $$100\times p\%$$ of the time.
Suppose $$M_1$$ is the statement that $$0.49 and $$M_2$$ says $$0.2 Then one must have $$\Pr(M_1)\le \Pr(M_2).$$
The probability, given the value of $$p,$$ of the outcome $$HH$$ is $$\Pr(HH\mid p) = p^2.$$ If that is regarded as a function of $$p,$$ then it is what is called a likelihood function: $$L(p) = p^2.$$
Suppose $$f(r)\, dr,$$ for $$0 is the probability distribution of $$p.$$ That implies, for example, that $$\displaystyle \Pr(0.3 and similarly for other intervals. Then $$\Pr(0.3 where $$c$$ is the normalizing constant, which satisfies $$c\int_0^1 r^2 f(r)\, dr = 1.$$
• Allright. So I understand the $Pr(M)$. But what about $Pr(M|D)$? You clarify the condtional expectation of the outcome HH very good. But could you clarify what we mean by the conditional expectation of a statement/model? – LocalMartingale Oct 25 '20 at 12:47