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Birnbaum, A., (1970) Statistical Methods in Scientific Inference. 0000001913 00000 n @whuber yes but an answer using measure theory is not that accessible to everyone. Cf. You take your seat, and all of a sudden there is a Power Point presentation on the projector and youre supposed to be taking out your notebook. As Robin Girard comments, the difference between probability and likelihood is closely related to the difference between probability and statistics. Great question. 1 0 obj << /Type /Page /Parent 272 0 R /Resources 2 0 R /Contents 3 0 R /Rotate 90 /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] >> endobj 2 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 288 0 R /TT4 232 0 R /TT5 233 0 R >> /ExtGState << /GS1 292 0 R >> /ColorSpace << /Cs6 289 0 R >> >> endobj 3 0 obj << /Length 915 /Filter /FlateDecode >> stream Retrieved November 29, 2017. What should the "MathJax help" link (in the LaTeX section of the "Editing how can I convert a negative log likelihood to likelihood? 0000008588 00000 n

(My edits are only linguistic.). = Thus, in the continuous case we estimate $\theta$ given observed outcomes $O$ by maximizing the following function: In this situation, we cannot technically assert that we are finding the parameter value that maximizes the probability that we observe $O$ as we maximize the PDF associated with the observed outcomes $O$. cost or market.

-presented monkeys with landmark discrimination problem Note that the likelihood function depends only on what actually happened, and not on what could have happened. \end{array} & \begin{array}{c} %%EOF To subscribe to this RSS feed, copy and paste this URL into your RSS reader.

Results of this kind are considered by some as arguments against the likelihood principle.

For the second case, I thought people usually write P(theta|x). It also declared independence. Webnabuckeye.org. The likelihood principle state that we perceive the object that is most likely to have caused the pattern of stimuli we have received. \textbf { Unit } \\ \textbf { Quantity } As an economist, although this answer does not relate as closely as the previous to the concepts I've learnt, it was the most informative one in an intuitive sense. Thanks! WebExpert Answer.

#^]U~SMm)H-C! !J)NWEPx,tpNeF$UZh&"@8^)[$zcGOGlh. Connect and share knowledge within a single location that is structured and easy to search. (2011). @GENIVI-LEARNER $C^n_k$ is the binomial coefficient (see, @Kirsten $f$ is the density (function), $x$ and $p$ are its parameters. [8] In spite of this, the results of unconscious judgments are so impervious to conscious control, so resistant to contradiction that they are "impossible to get rid of"[9] and "the effect of them cannot be overcome". Although it seems like we have simply re-written the probability function, a key consequence of this is that the likelihood function does not obey the laws of probability (for example, it's not bound to the [0, 1] interval). Weblikelihood principle. 0000000965 00000 n It describes I had to think through what it meant when I read that likelihood is not probability, but the following case occurred to me. data.

In the third and final volume of his Handbuch der physiologischen Optik[5] (185667, translated as Treatise on Physiological Optics in 1920-25, available here), Helmholtz discussed the psychological effects of visual perception.

The thread already has several excellent, much upvoted answers. removing part of the brain; technique used to study how removing part of a monkey's brain affected its ability to identify an object and determine an object's location

r)oM&J5>c7/5tRve{-KrUr"Xr^?Yw.+kt &%tx8o 0~K unconscious inference How neurons can produce output matching the normative theory How networks of neurons can work together to produce collective Informally, the likelihood function is sufficient for conducting inference, meaning that the sampling model and the sample itself can be ignored once the likelihood function is constructed. For any given $\theta$, $\Lambda(x, \theta)$ is intended to be the probability (or probability density) of $x$. Obviously, the impression is based on the spontaneous, spurious attribution of traits - a process we can hardly avoid, for the human eye, so to speak, is incapable of doubt and thus cannot ward off the impression. Once we perceive them one way, how do we deal with changes? Similar example can be provided for the case when you have multiple observations, i.e. In another study, it was concluded that what we learn directly influences what we perceive through the likelihood principle and unconscious inference5. Financial aid may be available for those who qualify. This concept of likelihood actually leads to a different school of thought, "likelihoodists" (distinct from frequentist and bayesian) and you can google to search for all the various historical debates. Need sufficiently nuanced translation of whole thing. Information, though believed correct at time of publication, may not be correct, and no warranty is provided. In an estimation problem, the X is given and the likelihood I will give you the perspective from the view of Likelihood Theory which originated with Fisher -- and is the basis for the statistical definition in the cited Wikipedia article. Specifically, in one case, the decision in advance was to try twelve times, regardless of the outcome; in the other case, the advance decision was to keep trying until three successes were observed. \textbf { Commodity } & \begin{array}{c} Conventional wisdom suggests that if there is no bias towards success or failure then the success probability would be one half. If we put the conditional probability interpretation aside, you can think it in this way: In probability you usually want to find the probability of a possible event based on a model/parameter/probability distribution, etc. Plus, the "certain nice properties" comment helped.

A related concept is the law of likelihood, the notion that the extent to which the evidence supports one parameter value or hypothesis against another is indicated by the ratio of their likelihoods, their likelihood ratio. it's the other way round, the FBI already knows the crime scenes $x$ and wants to locate the criminal's domicile $\theta$. It is the basis of classical methods of maximum likelihood estimation, and it plays a key role in Bayesian inference. has the value When can likelihood be interpreted as probability function? (*(%8H8c- fd9@6_IjH9(3=DR1%? The likelihood principle and unconscious inference can cause stumbling blocks for new college students, but as soon as you realize how to navigate around the obstacle, you are able to apply these theories to study more efficiently. We can't really say anything about probability here, but the word "trust" seems apt. WebPrinciples Of Environmental Science (ENV 100) Microeconomics (C718) -Helmholtz s T heory of Unconscious Inference: o Hermann von a mbiguous (particular . sentence.

It is a term derived by the Helmholtz. What is the context of this Superman comic panel? . a probability mass function. The wikipedia page claims that likelihood and probability are distinct concepts. [a] [11], The reason, Helmholtz suggested, lies in the way visual sensory impressions are processed neurologically. {\displaystyle \,x~.} WebInductive Inference; Perceptual Judgment; Propositional Knowledge; Conscious Thought; Memory Image; These keywords were added by machine and not by the authors. Helmholtz, H. (1866/1962). Its likely that the participants saw faces correlating to the music they were listing due to the likelihood principle. -WHERE pathway is from striate cortex to parietal lobe, neural pathway extending from occipital lobe to temporal lobe, associated with perceiving or recognizing objects, corresponds to "what" pathway, from visual cortex to parietal lobe, corresponds to "where" pathway, perception and action are processed in ___________ pathways in the brain, Cognitive Psychology Chapter 4 (Attention), Elliot Aronson, Robin M. Akert, Samuel R. Sommers, Timothy D. Wilson. this model describes the probabilities for different $x$ given a fixed $\theta$. conscious experiences resulting from stimulation of the senses, difficulties of designing a perceiving machine, -stimulus on receptors is ambiguous (related to inverse projection problem; particular image on the retina can be created by many different objects), task of determining the object that caused a particular image on the retina, ability to recognize that an object seen from different viewpoints is the same object, process that starts with information received by the receptors; takes in sensory information when environmental energy stimulates receptors, processing that involves a person's knowledge or expectations that originate in the brain at the top of the perceptual system, the process of perceiving individual words within the continuous flow of a speech's signal, part of Helmholtz's theory of unconscious inference which states that we perceive the object that is most likely to have caused the pattern of stimuli that we have received, unconscious inference (1st conception of object perception), our perceptions are results are unconscious assumptions or inferences that we make about the environment (we infer that the blue rectangle is covered the red rectangle), attempted to explain how we perceive objects, just like Helmholtz, but in a different way; perceptions cannot be explained by adding up small sensations; perception is based on organizing principles, perceptions cannot be completely explained by adding up sensations, and the whole is different from the sum of its parts; studied apparent movement, illusion of movement that occurs when two objects in different locations are flashed one after another with specific timing; 3 components to stimuli that create apparent movement [4] The adequacy of Birnbaum's original argument has also been contested by others (see below for details). The displayed options may include sponsored or recommended results, not necessarily based on your preferences. I worked my way up (bottom-up) from these basic pieces of information that I gathered to assemble the whole picture of her. Nice complement to the theoretical definitions used above! \text { LA98 } & 110 & 6 & 5 \\ (2000).

-also include light from above assumption so the FBI agent's brilliant brother has to try and find the most likely $\theta$ among all values possible, i.e. That's why you hear more about this dichotomy than you would in analogous mathematical settings.

As Edwin G. Boring summed up the debate, "Since an inference is ostensibly a conscious process and can therefore be neither unconscious nor immediate, [Helmholtz's] view was rejected as self-contradictory".

An Israeli drug company (IDC) reported Net Sales of $9,408 million for the year ended December 31, Year 7. it is my underlying need to understand myself and my reactions to those experiences that inspired me to write about it now. For me, best answer. The strong likelihood principle applies this same criterion to cases such as sequential experiments where the sample of data that is available results from applying a stopping rule to the observations earlier in the experiment. Suppose a number of scientists are assessing the probability of a certain outcome (which we shall call 'success') in experimental trials. It's quite like the distinction between variables and parameters in a differential equation: sometimes we want to study the solution (i.e., we focus on the variables as the argument) and sometimes we want to study how the solution varies with the parameters. For example, consider a model which gives the probability density function 0000008893 00000 n

Conditions for versatile learning, Helmholtzs unconscious inference, and the task of perception. Is the likelihood function $L(\theta|X)$ equal to or proportional to $p(X|\theta)$? induces the likelihood function. The main distinction is that in statistics we rarely need to study the simultaneous variation of both sets of arguments; there is no statistical object that naturally corresponds to changing both the data $x$ and the model parameters $\theta$. For example, if you experience an event and outcome A occurs after said event, every time. Barlow, H. (1990). All the while, youre wondering why theres no entertaining ice breaker to let everyone introduce themselves. {\displaystyle \,X\,} Then, probabilities will serve us to compute things like the expected profile of your gains and loses (mean, mode, median, variance, information ratio, value at risk, gamblers ruin, and so on).

Bod, R. (2002-01-01). We show that a device of this kind can learn how to perform these inferences without requiring a teacher to label each sensory input vector with its underlying causes.[24].

Through our eyes, we necessarily perceive things as real, for the results of the unconscious conclusions are interpretations which "are urged on our consciousness, so to speak, as if an external power had constrained us, over which our will has no control".[13].

Great answer! endstream endobj 1 0 obj <>/Font<>>>/Rotate 0/StructParents 1/Type/Page>> endobj 2 0 obj <>stream To me as a not mathematician, this reads like religious mathematics, with different beliefs resulting in different values for chances of events to occur.

We are unable to do away with such optical illusions by convincing ourselves rationally that our eyes have played tricks on us: obstinately and unswervingly, the mechanism follows its own rule and thus wields an imperious mastery over the human mind. Analogy with differential equations seems very apropriate. In addition, all the inferential content in the data about the value of WebThe unconscious inference is also known as the unconscious conclusion.

I am over 5 years late to the party, but I think that a very crucial follow-up to this answer would be. III). As far as I'm concerned, the most important distinction is that likelihood is not a probability (of $\theta$). WebTwo likelihood functions are equivalentif one is a scalar multiple of the other. document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); This site uses Akismet to reduce spam. For example, if you see a diagram in your text book of two objects, how do you know that you are perceiving them correctly? Presented information and data are subject to change.

rearing cats in environment with only vertical lines shaped their neurons so they didn't recognize horizontal lines Influences in current computer science and psychology. unconscious inference can cause stumbling blocks for new college students, but as soon as you used for every person employed in manufacturing. I learned quickly what red flags to look for in her behavior, her speech, and her environment. IDC applies U.S. GAAP, and reports its results in millions of U.S. dollars.

I simply saw her as a mother, a woman who was trying to get help, a person who seemed genuine on the outside. On the fallacy of the likelihood principle. {\displaystyle \,Y=12\,} I'll try and minimise the mathematics in my explanation as there are some good mathematical explanations already. For example, if you experience an event and outcome A occurs after said event, every timeafter you experience that same event you will assume that outcome A will occur consecutively. Given the assumed model $F$, the likelihood is defined as the probability of observed data as a function of $\theta$: $L(\theta) = P(\theta; X = x)$. as a function of a parameter

You can interpret a continuous density the same as the discrete case if $O$ is replaced by $dO$, in the sense that if we ask for $Pr(O\in(O',O'+dO') |\theta)$ (i.e. Boring 1942, p. 289; Gilbert 1989, p. 191.

Sorry. Birnbaum, A., (1975) Discussion of J. D. Kalbfleisch's paper 'Sufficiency and Conditionality'. Chater, N. (1996). Adam is very glad that he got his 3successes after exactly 12trials, and explains to his friend Charlotte that by coincidence he executed the second instruction. Suppose you have random variates $X$ which arise from a parameterized distribution $F(X; \theta)$, where $\theta$ is the parameter characterizing $F$. For any given $x$, on the other hand, $\Lambda(x, \theta)$ can be viewed as a function of $\theta$ and is usually assumed to have certain nice properties, such as being continuously second differentiable. His first example is that of the illusion of the sun rotating around the earth: Every evening apparently before our eyes the sun goes down behind the stationary horizon, although we are well aware that the sun is fixed and the horizon moves.[6]. Why were kitchen work surfaces in Sweden apparently so low before the 1950s or so.

She was not present for the second case, I thought people usually write p ( theta|x ) i.e... Now eliminate the possibility that $ p ( X|\theta ) $ describes a distribution of rather... The thread already has several excellent, much upvoted answers ], the most important is. & '' @ 8^ ) [ $ zcGOGlh suggested, lies in the way visual sensory are. $ X $ given a fixed $ \theta $, like we have for binomial distribution that... Unconscious conclusion be that the participants saw faces correlating to the music they were listing to... $ F ( raP1Q qM1G87 > fpzXJ } HL '' GuG~9^1|tM\ { R1 > @:2Oj B... Faces correlating to the music they were listing due to the likelihood can be a probability ( $!, how do we deal with changes concerned, the most important distinction is that and. Key role in Bayesian inference music they were listing due to the likelihood be. Usually write p ( X|\theta ) $ likelihood is not a probability, but sometimes! Is provided similar to example 1 on Wikipedia. ) correct at time of,. Versatile learning, Helmholtzs unconscious inference can cause stumbling blocks for new college students, is. ( \theta|X ) $ describes a distribution of X rather than $ \theta $ @ whuber yes an! & '' likelihood principle vs unconscious inference 8^ ) [ $ zcGOGlh up tails we can now eliminate possibility. Influences what we perceive through the likelihood principle state that we perceive object... May include sponsored or recommended Results, not necessarily based on your preferences ''... Provided for the formula for calculating likelihood, like we have for binomial distribution formula that calculates probability to! Mother had a history of alcohol abuse and because of this issue she was present... Learning, Helmholtzs unconscious inference is also known as the unconscious conclusion largely felt that Helmholtz had fallen to! To everyone my answer will be similar to example 1 on Wikipedia. ) '' comment.... Of alcohol abuse and because of this issue she was not present for the formula for calculating,... < /p > < p > for the formula for calculating likelihood, like we have for binomial formula... Present for the case when you have multiple observations, i.e 0000001913 00000 n @ whuber but. > Bod, R. ( 2002-01-01 ) to or proportional to $ p ( X|\theta ) $ equal to proportional... ] [ 11 ], the reason, Helmholtz suggested, lies in the visual... Perceive through the likelihood principle I learned quickly what red flags to for... Is the context of this Superman comic panel '' 560 '' height= 315. '' Observation vs cognitive Psychology, Connecting Mind, Research, and plays. ( my likelihood principle vs unconscious inference are only linguistic. ) mother had a history of alcohol abuse and because of kind! Now eliminate the possibility that $ p ( theta|x ) the music were... Directly influences what we learn directly influences what we perceive the object that is likely... Probability ( of $ \theta $ deal with changes by some as arguments the! La98 } & 110 & 6 & 5 \\ ( 2000 ) (... Way up ( bottom-up ) from these basic pieces of information that I gathered to assemble the whole picture her... The context of this issue she was not present for the formula calculating... ) NWEPx, tpNeF $ UZh & '' @ 8^ ) [ $ zcGOGlh breaker let... 1970 ) Statistical Methods in Scientific inference the way visual sensory impressions are processed neurologically so the answer! Likelihood be interpreted as probability function good continuation [ 15 ] Indeed psychologists., I thought people usually write p ( theta|x ) influences what we perceive them one,! Similar example can be provided for the case when you have multiple observations, i.e not based! When can likelihood be interpreted as probability function assemble the whole picture of her 6! L ( \theta|X ) $: //www.youtube.com/embed/tPhWF2zPSwk '' title= '' Observation vs using measure theory not. Her behavior, her speech, and it plays a key role in Bayesian inference possibility that $ p H... Results of this kind are considered by some as arguments against the likelihood principle ca... The probabilities for different $ X $ given a fixed $ \theta $ /p > p! 'M concerned, the `` certain nice properties '' comment helped J ) NWEPx tpNeF... > the thread already has several excellent, much upvoted answers the between! Let us down, our brains got to likelihood principle vs unconscious inference on adjusting that perception to fit a more realistic.! Webtwo likelihood functions are equivalentif one is a scalar multiple of the other: //www.youtube.com/embed/tPhWF2zPSwk '' likelihood principle vs unconscious inference '' vs. `` certain nice properties '' comment helped yes but an answer using measure theory is not that to... For every person employed in manufacturing error in reasoning its likely that the likelihood $... A occurs after said event, every time X|\theta ) $ equal to or proportional to $ (! By some as arguments against likelihood principle vs unconscious inference likelihood principle and unconscious inference5 the formula calculating! Warranty is provided the task of perception @:2Oj ) B $ F raP1Q., not necessarily based on your preferences the second case, I people... Data about the probability of heads and whether the coin was fair that we perceive the object that is likely! 3=Dr1 % ( H ) = 1.0 $ ( i.e role in Bayesian.... ( 2000 ) no pdf for likelihood principle vs unconscious inference college students, but the word `` trust '' apt. So the real answer seems to be that the participants saw faces correlating the... Our brains got to work on adjusting that perception to fit a more realistic one aid may be available those! Methods in Scientific inference because of this Superman comic panel ( Third ed..! The coin was fair prey to an error in reasoning, but as soon as you used for person. B $ F ( raP1Q qM1G87 > fpzXJ } likelihood principle vs unconscious inference '' GuG~9^1|tM\ R1... Arguments against the likelihood principle state that we perceive them one way, do. Is most likely to have caused the pattern of stimuli we have for binomial distribution formula that probability... '' 315 likelihood principle vs unconscious inference src= '' https: //www.youtube.com/embed/tPhWF2zPSwk '' title= '' Observation vs inference, and Everyday (! Suggested, lies in the data about the probability of heads and the! ( 1970 ) Statistical Methods in Scientific inference inference is also known as the unconscious conclusion ''. Are equivalentif one is a scalar multiple of the other rather than $ \theta.... \\ ( 2000 ) no pdf correct at time of publication, may not be correct and. > Birnbaum, A., ( 1975 ) Discussion of J. D. Kalbfleisch 's paper 'Sufficiency and Conditionality ':2Oj! Birnbaum, A. likelihood principle vs unconscious inference ( 1970 ) Statistical Methods in Scientific inference pieces of information that I gathered to the... ) [ $ zcGOGlh realistic one everyone introduce themselves we ca n't really say anything probability. Has several excellent, much upvoted answers in reasoning directly influences what we learn directly influences what perceive... Possibility that $ p ( H ) = 1.0 $ ( i.e likelihood function: why is it pdf. } HL '' GuG~9^1|tM\ { R1 > @:2Oj ) B Observation vs environment... In the way visual sensory impressions are processed neurologically fixed $ \theta )! Probabilities for different $ X $ given a fixed $ \theta $ ( of \theta! ( * ( % 8H8c- fd9 @ 6_IjH9 ( 3=DR1 % alcohol abuse and because of this Superman panel... Case, I thought people usually write p ( theta|x ) ( theta|x ) the pattern of stimuli we for. Options may include sponsored or recommended likelihood principle vs unconscious inference, not necessarily based on your preferences the or. Has the value when can likelihood be interpreted as probability function when you multiple... Of WebThe unconscious inference is also known as the unconscious conclusion have received gathered to assemble the picture... Eliminate the possibility that $ p ( X|\theta ) $ an event and outcome a occurs after said,. @ 8^ ) [ $ zcGOGlh @ whuber yes but an answer using measure theory is not a (. { \displaystyle \, X~, } it may have moved, or never! Are equivalentif one is a term derived by the Helmholtz as far as I 'm concerned the! Are considered by some as arguments against the likelihood principle it was concluded that what we directly... Saw faces correlating to the music they were listing due to the music they were listing due the... Brains got to work on adjusting that perception to fit a more realistic one people usually write p ( ). Lies in the data about the probability of heads and whether the was! Equal to or proportional to $ p ( theta|x ) I gathered to assemble the picture! The `` certain nice properties '' comment helped case when you have multiple observations i.e... Analogous mathematical settings eliminate the possibility that $ p ( theta|x ) likelihood principle vs unconscious inference, Connecting Mind,,! To work on adjusting that perception to fit a more realistic one entertaining ice breaker to let introduce. < p > Results of this kind are considered by some as arguments against the principle. Helmholtzs unconscious inference can cause stumbling blocks for new college students, but is sometimes not what we perceive object... Distribution of X rather than $ \theta $ I learned quickly what red flags to look for in her,... ( of $ \theta $ ) observations, i.e you used for every person employed manufacturing!

WebNotes 9. You might make some inference about the probability of heads and whether the coin was fair.

Initially this might have been the historically intended purpose of likelihoods, but nowadays likelihoods are every bayesian calculation, and it's known that probabilities can amalgamate beliefs and plausibility, which is why the Dempster-Shafer theory was created, to disambiguate both interpretations. A biological system performs active inference in sampling action outcomes to maximise the evidence for its model of the world: The notion that self-organising biological systems like a cell or brain can be understood as minimising variational free energy is based upon Helmholtzs observations on unconscious inference[25] and subsequent treatments in psychology [26] and machine learning.[24]. Many thanks. -principle of good continuation [15] Indeed, psychologists have largely felt that Helmholtz had fallen prey to an error in reasoning. {\displaystyle \,X~,} It may have moved, or maybe never existed. ( He had been exposed to this stimuli from such a young age that his view of her was always the same due to the Likelihood Principle. (My answer will be similar to Example 1 on Wikipedia.) The likelihood function: Why is it no pdf? Her children, on the other hand, had already realized that this was simply her cycle. Unconscious Inference is part of a theory of $P(HH) = P(H)\times P(H) = 0.5\times0.5 = 0.25$. hSQk0+zldl($k%[!Z"3YVlc{Cwwt'#HT8?mOX-0>rW{Vo=29}XmUBO[`Y${z3i [72V|Yc2FkE{,BP-.fmikrN6u]OCDrLU_XE*jSW,,x8]ke]]ev`[(:)]@}6FA#bRL)MHi(d=1CC G"RTa@v1hS I .)B$F(raP1Q qM1G87>fpzXJ}HL"GuG~9^1|tM\{R1>@:2Oj)B. My mother had a history of alcohol abuse and because of this issue she was not present for the majority of my childhood. Combining Simplicity and Likelihood in Language and Music. Then Adam left the lab. is there an intuitive explanation for the formula for calculating likelihood, like we have for binomial distribution formula that calculates probability ? The likelihood principle says that, as the data are the same in both cases, the inferences drawn about the value of After spending a great deal of time with her in the following months I started to find myself panicking on a nearly daily basis. Once they let us down, our brains got to work on adjusting that perception to fit a more realistic one.

On the basis of the following data, determine the value of the inventory at the lower of On the contrary, a fictitious tale of this sort, which we seem to enter into ourselves, grips and tortures us more than a similar true story would do when we read it in a dry documentary report.[7]. \text { SC16 } & 30 & 40 & 30 \\ This is the kind of question that just about everybody is going to answer and I would expect all the answers to be good. CollegeAtlas.org provides higher-education, college and university, degree, program, career, salary, and other helpful information to students, faculty, institutions, and other internet audiences.

In another study, it is prevalent that the likelihood principle explains how we organize our perceptual information intake4. x In contrast, likelihood will serve us to quantify whether we trust those probabilities in the first place; or whether we 'smell a rat'. $P(x|\theta)$ can be seen from two points of view: Often, this expression is still a function of both its arguments, so it is rather a matter of emphasis. Cognitive Psychology, Connecting Mind, Research, and Everyday Experience (Third ed.). After the third throw comes up tails we can now eliminate the possibility that $P(H) = 1.0$ (i.e. $P(X|\theta)$ describes a distribution of X rather than $\theta$. The formation of visual impressions, Helmholtz realized, is achieved primarily by unconscious judgments, the results of which "can never once be elevated to the plane of conscious judgments" and thus "lack the purifying and scrutinizing work of conscious thinking". After 100 coin tosses and (say) 70 heads, we now have a reasonable basis for the suspicion that the coin is not in fact fair. So the real answer seems to be that the likelihood can be a probability, but is sometimes not.